8]<OU_1 60532 >m
OSMANIA UNIVERSITY LIBRARY
Call No. 5/7'5g//f &7 J) Accession No. V> * 7 2--
Author /[ . J.
Title (!),U 7
y llLi'Lt \^ (^
This BooliMiouId be returned on or before the date last marked below*
AMERICAN MATHEMATICAL SOCIETY
COLLOQUIUM PUBLICATIONS, VOLUME XIV
DIFFERENTIAL EQUATIONS FROM THE
ALGEBRAIC STANDPOINT
BY
JOSEPH PELS RITT
PROFESSOR OF MATHEMATICS
COLUMBIA UNIVERSITY
NEW YORK
PUBLISHED BY THE
AMERICAN MATHEMATICAL SOCIETY
501 WEST 116TH STREET
1932
Photo-Lithoprint Reproduction
KDWARDS BROTHERS, INC.
Lithoprinters
ANN ARBOR, MICHIGAN
1947
INTRODUCTION
We shall be concerned, in this monograph, with systems of
differential equations, ordinary or partial, which are algebraic
in the unknowns and their derivatives. The algebraic side
of the theory of such systems seems to have remained, up
to the present, in an undeveloped state.
It has been customary, in dealing with systems of differential
equations, to assume canonical forms for the systems. Such
forms are inadequate for the representation of general systems.
It is true that methods have been proposed for the reduction
of general systems to various canonical types. But the limit-
ations which go with the use of the implicit function theorem,
the lack of methods for coping with the phenomena of
degeneration which are ever likely to occur in elimination
processes and the absence of a technique for preventing the
entrance of extraneous solutions, are merely symptoms of the
futility inherent in such methods of reduction.
Now, in the theory of systems of algebraic equations, one
witnesses a more enlivening spectacle. Kronecker's Fest-
schrift of 1882 set upon a firm foundation the theory of
algebraic elimination and the general theory of algebraic
manifolds. The contributions of Mertens, Hilbert, KOnig,
Lasker, Macaulay, Henzelt, Emmy Noether, van der Waerden
and others, have brought, to this division of algebra, a high
degree of perfection. In the notions of irreducible manifold,
and polynomial ideal, there has been material for far reaching
qualitative and combinatorial investigations. On the formal
side, one has universally valid methods of elimination and
formulas for resultants.
To bring to the theory of systems of differential equations
which are algebraic in the unknowns and their derivatives,
iii *
IV INTRODUCTION
some of the completeness enjoyed by the theory of systems
of algebraic equations, is the aim of the present monograph.
The point of view which we take is that of our paper
Manifolds of functions defined by systems of algebraic differ-
ential equations, published in volume 32 of the Transactions
of the American Mathematical Society. In what follows,
we shall outline our results.
Chapters I- VIII treat ordinary differential equations. We
deal with any finite or infinite system of algebraic differential
equations in the unknown functions y ly , y n of the variable x .
We write each equation in the form
F(x\ y l9 - -,y n ) = 0,
where F is a polynomial in the yi and any number of their
derivatives. The coefficients in F will be supposed to be
functions of x, meromorphic in a given open region. An
expression like F, above, will be called a form. All forms
considered in this introduction will be understood to have
coefficients which are contained in a given field. By a field,
we mean a set of functions which is closed with respect to
rational operations and differentiation.*
Let ~ be any finite or infinite system of forms in y^, - -, y n >
By a solution of 2, we mean a solution of the system of
equations obtained by setting the forms of 2 equal to zero.
The totality of solutions of 2 will be called the manifold
of 2. If ^\ and 2 2 are systems such that every solution
of 2 l is a solution of 2, we shall say that 2 2 holds 2 l9
A system - will be called reducible or irreducible according
as there do or do not exist two forms, and H, such that
neither G nor H holds 2, while GH holds 2. The manifold
of 2, and also the system of equations obtained by equating
the forms of 2 to zero, will be called reducible or irreducible
according as 2 is reducible or irreducible.
We can now state the principal result of Chapter I. Every
manifold is composed of a finite number of irreducible manifolds.
*A formal definition is given in 1.
INTRODUCTION V
That is, given any system 2, there exist a finite number of
irreducible systems, 2 l9 - -, 2 S , such that 2 holds every 2,,
while every solution of -5" is a solution of some 2,. The
decomposition into irreducible manifolds is essentially unique.
Let us consider an example. The equation
> ()'-*' -
whose solutions are y (x a) 2 , (a constant), and y = 0,
is a reducible system in the field of all constants. For
(9} 9
\ A) ~i 1 7 9 &
ax \ dx~
holds the first member of (1), while neither factor in (2)
does. The system (1) is equivalent to the two irreducible
systems
-4y = 0, - =
dxl J dx
and
The decomposition theorem follows from a lemma which
bears a certain analogy to Hilbert's theorem on the existence
of a finite basis for an infinite system of polynomials. We
prove that if 2 is an infinite system of forms in y\ , , y nj
then 2 contains a finite subsystem whose manifold is identical
with that of 2.*
Chapters II and VI study irreducible manifolds. We start,
in Chapter II, with a precise formulation of the notion of
general solution of a differential equation. We do not think
that such a formulation has been attempted before. Let A
be a form in y iy -, y n , effectively involving y n , and irre-
ducible, in the given field, as a polynomial in the yi and
* See 124 for a comparison, with a theorem of Tresse, of the ex-
tension of this lemma to partial differential equations.
yi INTRODUCTION
their derivatives. Let the order of the highest derivative
of y n in A be r and let y w represent that derivative. Let
2 be the totality of forms which vanish for all solutions of
A with dA/dy nr ^O. We prove that 2 is irreducible. The
manifold of 2 is one of the irreducible manifolds in the
decomposition of the manifold of A . We call this manifold
the general solution of A (or of A = 0).
The remainder of Chapter II deals with the association,
with every irreducible system -2", of a differential equation
which we call a resolvent of 2. The first member of the
resolvent is an irreducible polynomial, so that the resolvent
has a general solution. Roughly speaking, the determination
of the general solution of the resolvent is equivalent to the
determination of the manifold of 2. The theory of resol-
vents furnishes a theoretical method for the construction of
all irreducible systems. One will see that the resolvent can
be used advantageously in formal problems.
In Chapter VI, we study what might be called the texture
of an irreducible manifold. For the case of the general
solution of an algebraically irreducible form, our work amounts
to characterizing those singular solutions (solutions with
9 A 1 3 y nr = 0) which belong to the general solution.
Chapters V and VII contain, among other results, finite
algorithms, involving differentiations and rational operations,
for decomposing a finite system into irreducible systems and
for constructing resolvents. In Chapter V, we do not obtain
the actual irreducible systems, but rather certain basic sets
of forms (Ch. II) which characterize the irreducible systems.
However, this permits the construction of resolvents. In
Chapter VII, a process is obtained which, if carried sufficiently
far, will actually produce the irreducible systems. Unfortunately,
there is nothing in this process which informs one, at any
point, as to whether or not the process has had its desired effect.
The results of Chapter V furnish a complete elimination
theory for systems of algebraic differential equations.
In Chapter VII, we derive an analogue, for differential
forms, of the famous Nullstellensatz of Hilbert and Netto. In
INTRODUCTION VU
Chapter VIII, we present an analogue of Liiroth's theorem
on the parameterization of unicursal curves. In Chapter III,
there will be found a theory of resultants of pairs of differ-
ential forms. A number of other special results are distributed
through the monograph.
In Chapter X, some of the main results stated above are
extended to systems of algebraic partial differential equations.
In particular, an elimination theory is obtained for such
systems.
Chapter IV treats systems of algebraic equations. The chief
purpose is to obtain special theorems, and finite algorithms,
for application to differential equation theory. The main
results of Chapter IV are known ones, but the treatment
appears new, and some special theorems, of importance for
us, do not seem to exist in the literature.
It has been our aim to give this monograph an elementary
character, and to assume only such facts of algebra and
analysis as are contained in standard treatises. With this
principle in mind, we have devoted Chapter IX to an
exposition of Riquier's remarkable existence theorem for
orthonomic systems of partial differential equations.
Thus Chapter IX is purely expository, and Chapter IV is
largely so. The remaining chapters present results contained
in our above mentioned paper, and results communicated by
us to the American Mathematical Society since the publication
of that paper.
Koenigsberger's irreducible differential equations,* and
Drach's irreducible systems of partial differential equations,t
are irreducible in the sense described above. In Drach's
definition, which includes that of Koenigsberger, a system
is called irreducible if every equation which admits one
solution of the system admits all solutions of the system.
Thus, systems which are irreducible in our sense may easily
be reducible in the theories of Koenigsberger and Drach.
The definitions of Koenigsberger and Drach, which do not
* Lehrbuch der Differenzialgleichungen, Leipzig, 1889.
t Annales de TEcole Normale, vol. 34, (1898).
Vlli INTRODUCTION
lead to decompositions into irreducible systems, are the
starting points of group-theoretic investigations, which parallel
the Galois theory. Our course, as we have seen, is in a
different direction.
Many questions still remain for investigation. In particular,
a theory of ideals of differential forms and a theory of
birational transformations, await development.* Chapters VII
and VIII may perhaps be regarded as rudimentary beginnings
of such theories.
It goes without saying that we have been guided, in our
work, by the existing theory of algebraic manifolds. We
have found particularly valuable, the excellent treatment of
systems of algebraic equations given in Professor van der
Waerden's paper Zur Nullstellentheorie der PolynomidealeA
But it is not surprising, on the other hand, that the investigation
of essentially new phenomena should have called for the
development of new methods.
I am very grateful to the Colloquium Committee of the
American Mathematical Society, who have invited me to
lecture on the subject of this monograph at the University
of California in September, 1932. To my friend and colleague
Dr. Eli Gourin, who assisted me in reading the proofs, I extend
my deep thanks.
* In connection with transformations of general (non- algebraic) differ-
ential equations, see Hilbert, Mathematische Annalen, vol. 73 (1913), p. 95.
t Mathematische Annalen, vol. 96, (1927), p. 183.
NEW YORK, N. Y.
February, 1932.
J. F. Rrrr.
CONTENTS
Page
CHAPTER I
DECOMPOSITION OF A SYSTEM OF ORDINARY ALGEBRAIC
DIFFERENTIAL EQUATIONS INTO IRREDUCIBLE SYSTEMS . 1
Fields, forms, ascending sets, basic sets, reduction, solutions
and manifolds, completeness of infinite systems, non-existence of
a Hilbert theorem, irreducible systems, the fundamental theorem,
uniqueness of decomposition, examples, relative reducibility, ad-
junction of new unknowns, fields of constants.
CHAPTER II
GENERAL SOLUTIONS AND RESOLVENTS 21
General solution of a differential equation, closed systems,
arbitrary unknowns, the resolvent, invariance of the integer g,
order of the resolvent, construction of irreducible systems, irre-
ducibility and the open region 21.
CHAPTER III
FIRST APPLICATIONS OF THE GENERAL THEORY 47
Resultants of differential forms, analogue of an algebraic theo-
rem of Kronecker, form quotients.
CHAPTER IV
SYSTEMS OF ALGEBRAIC EQUATIONS 62
Indecomposable systems of simple forms, simple resolvents,
basic sets of prime systems, construction of resolvents, resolution
of a finite system into indecomposable systems, a special theorem.
CHAPTER V
CONSTRUCTIVE METHODS 92
Characterization of basic sets of irreducible systems, basic sets
in a resolution of a finite system into irreducible systems, test for
a form to hold a finite system, construction of resolvents, a remark
on the fundamental theorem, Jacobi-Weierstrass canonical form.
ix
X CONTENTS
Page
CHAPTER VI
CONSTITUTION OF AN IRREDUCIBLE MANIFOLD 100
Seminorraal solutions, adjunction of new functions to o?, inde-
composability and irreducibility.
CHAPTER VII
ANALOGUE OF THE HILBERT-NETTO THEOREM. THEORETICAL
DECOMPOSITION PROCESS . . 108
Analogue of Hilbert-Netto theorem; theoretical process for
decomposing a finite system of forms into irreducible systems;
forms in one unknown, of first order.
CHAPTER VIII
ANALOGUE FOR FORM QUOTIENTS OF MROTH'S THEOREM. 124
CHAPTER IX
RIQUIER'S EXISTENCE THEOREM FOR ORTHONOMIC SYSTEMS 135
Monomials, dissection of a Taylor series, marks, orthonomic
systems, passive orthonomic systems.
CHAPTER X
SYSTEMS OF ALGEBRAIC PARTIAL DIFFERENTIAL EQUATIONS 157
Decomposition of a system into irreducible systems, basic sets
of closed irreducible systems, algorithm for decomposition,
analogue of the Hilbert-Netto theorem.
CHAPTER I
DECOMPOSITION OF A SYSTEM OF ORDINARY ALGEBRAIC
DIFFERENTIAL EQUATIONS INTO
IRREDUCIBLE SYSTEMS
FIELDS
i. We consider functions meromorphic in a given open
region 21 in the plane of the complex variable x* We
recall that an open region is a set of points such that
(a) every point of the set is the center of a circle of positive
radius, all of whose points belong to the set;
(b) any two points of the set can be joined by a continuous
curve whose points all lie in the set.
A set r, of functions described as above, will be called
a field if
(a) & contains at least one function which is not identically zero;
(b) given any two functions f and g (distinct or equal), be-
longing to *& , then fiLg and fg belong to 7
(c) given any two functions, f and g y belonging to oF, ivith
g not identically zero, then fig belongs to 7
(d) given any function, f, in $r, the derivative of f belongs
to 3.
Every field contains all rational constants. Examples of
fields are: the totality of rational constants; the totality of
rational functions of &; all rational combinations of a: and e*
with constant coefficients; all elliptic functions with a given
period parallelogram.t
* We are dealing here only with the finite plane.
fThe notion of field of analytic functions has appeared previously,
among other places, in Picard's group-theoretic investigations on linear
1
2 ALGEBRAIC DIFFERENTIAL EQUATIONS
FORMS
2. In what follows, we work with an arbitrary field cF,
which is supposed to be assigned in advance and to stay
fixed.
We are going to develop some notions in preparation for
the study of differential equations in n unknown functions,
Vi> -->y n -
By a differential form or, more briefly, by a form, we
shall understand a polynomial in the y i and any number of
their derivatives, with coefficients meromorphic in 21.
With respect to every form introduced into our work, we
shall assume, unless the contrary is stated, that its coefficients
belong to S r .
Differentiation of functions y t will be indicated by means
of a second subscript. Thus
We write, frequently, y. = y io .*
Throughout cur work, capital italic letters will denote forms.
By the jth derivative of A, we mean the form obtained by
differentiating A f times with respect to x, regarding y t , , y n
as functions of x.
By the order of A with respect to yi, if A involves yi or
some of its derivatives effectively, we shall mean the greatest,;'
differential equations and in Landau's work on the factorization of linear
differential operators. See Picard, Traite d'Analyse, 2nd edition, vol. 3,
p. 562. The foregoing writers make the additional assumption that r
contains all constants. Loewy, however, in his work on systems of linear
differential equations, Mathematische Aiinalen, vol. 62 (1906), p. 89, does
not make this additional assumption. No generality would be gained by
allowing to consist of functions analytic except for isolated singularities.
With this assumption, it is an easy consequence of Picard's theorem on
essential singularities, and of the fact that r contains all rational constants,
that the functions in c7 are meromorphic.
* In certain problems, we shall use unsubscripted letters to represent
unknowns. If y is such an unknown, y^ will represent the jth derivative
of y.
I. IRREDUCIBLE SYSTEMS 3
such that yij is present in a term of A with a coefficient
distinct from zero. If A does not involve j/$, the order of A
with respect to yi will be taken as 0.
By the class of A, it A involves one or more yi effectively,
we shall mean the greatest p such that some y p j is effectively
present in A. If A is simply a function of x, A will be said
to be of class 0.
Let AI and A* be two forms. If A 2 is of higher order
than Ai in some y p , A 2 will be said to be of higher rank
than A l9 and A^ of lower rank than A 2 , in y p . If A and A 2
are of the same order, say q, in y p and if A 2 is of greater
degree than A l in y pq * then, again, A 2 will be said to be
of higher rank than A l in y p . Two forms for which no
difference in rank is established by the foregoing criteria
will be said to be of the same rank in y p .
If A 2 is of higher class than A i9 A 2 will be said to be of higher
rank than A\A If A 2 and AI are of same class p>0, and
if A 2 is of higher rank than A t in y p , then, again, A 2 will
be said to be of higher rank than AI. Two forms for which
no difference in rank is created by the preceding, will be
said to be of the same rank.|
If A 2 is higher than AI, A s higher than A 2j then A$ is
higher than A it
In later chapters, we shall have occasion to use other
symbols than y l9 -, y n for the unknowns. If the unknowns
are given in the order u,v, -, w, then, in the definitions
of class and of relative rank, the ^th unknown from the left
is to be treated like y p above.
We shall need the following lemma:
LEMMA. If
AI , A 2 , , Aq,
* Considered as a polynomial in y pq . If a form is identically zero (hence
of order zero in every y^ it will be considered of degree in every y p o.
This leads to no difficulties.
fWe shall frequently say, simply, "At is higher than A v ".
JThus, all forms of class zero are of the same rank.
4 ALGEBRAIC DIFFERENTIAL EQUATIONS
is an infinite sequence such that, for every q, Aq+i is not
higher than A q , there exists a subscript r, such that, for q>r,
A q has the same rank as A r .
The classes of the A q form a non-increasing set of non-
negative integers. It is then clear that, for q large, the
A q have the same class, say p. If p >0, the A q with q large
will be of the same order, say s, in y p . Finally, the Aq will
eventually have a common degree in y ps .
An immediate consequence of this lemma is that every
finite or infinite aggregate of forms contains a form ivhich is
not higher than any other form of the aggregate.
ASCENDING SETS
3. If AI is of class ^)>0, ^4.2 will be said to be reduced
with respect to A if A 2 is of lower rank than AI in y p .
The system
(1) A t , A 2 , > , A r
will be called an ascending set if either
(a) r = 1 and A l ^
or
(b) r >1, AI is of class greater than 0, and, for j>i, Aj is
of higher class than Ai and reduced ivith respect to Ai.
Of course, r <C n.
The ascending set (1) will be said to be of higher rank
than the ascending set
(2) J3 lf ft, ,,
if either
(a) There is a j. exceeding neither r nor s, such that Ai and
Bi are of the same rank for i < j and that Aj is higher
than BJ*
or
(b) s>r and Ai and Bi are of the same rank for i < r.
Two ascending sets for which no difference in rank is
created by what precedes will be said to be of the same
* If j = 1 , this is to mean that AI is higher than BI .
I. IRREDUCIBLE SYSTEMS 5
rank. For such sets, r = s and At and Bi are of the same
rank for every i.
Let <Z>!, <Z> 2 , tf> 3 be ascending sets such that <Z>i is higher
than <Z> 2 , O 2 higher than <D 3 . We write O^^, <P 2 ><Z> 3 .
We shall prove that <J>i > tf> 8
Let 0>i and tf> 2 be represented by (1) and (2) respectively
and let <Z> 8 be
Cl> /2 > * ' ' 7 Lt
Suppose first that <Z>i > 2 for the reason (a) and that
<Z>2 > #8 for the reason (a). Let j be the smallest integer
such that Bj is higher than Q-. Then either Ai is of the
same rank as BI for i < ./ or there is a k <^ j such that
.A t - is of the same rank as Bi for i < k but that ^ is higher
than Bk. In either case, Q>i > <P 8 by (a).
Suppose now that c^XD* by (b), while tf> 2 >0> 8 by (a).
Let y be taken as above. If j>r, <2>i > <*> 8 by (b). It j <r,
<Z>i><*> 8 by (a).
Now let 0*! > 0> 2 by (a), while 0> 2 > <Z> 8 by (b). Let j be
the smallest integer for which Aj is higher than Bj. Then
4/ is higher than Cj and Ai is of the same rank as d for
*<y. Thus Q> 1 >Q> 3 through (a).
Finally, if (D^^g by (b) and 0> 2 >a> 8 by (b), then
0>i > <Z>3 by (b).
We shall need the following fact:
Let
(3) <t> l9 0> 2 , -.., O q , ...
fe aw infinite sequence of ascending sets such that Q> q +i is not
higher than <Z> g for any q. Then there exists a subscript r
such that, for q>r, O q has the same rank as <Z> r .
By the lemma of 2, the first forms of the 0> q (A t in (1))
are all of the same rank for q large. This accounts for the
case in which <Z> g with q large has only one form. We may
thus limit ourselves to the case in which 0> q with q large
has at least two forms. The second forms will eventually
be of the same rank. Continuing, we find, since no <t> q has
more than n forms, that the <t> q with q large all have the
6 ALGEBRAIC DIFFERENTIAL EQUATIONS
same number of forms, corresponding forms being of the same
rank. This proves the lemma.
An immediate consequence of this result is that every finite
or infinite aggregate of ascending sets contains an ascending
set whose rank is not higher than that of any other ascending
set in the aggregate.
BASIC SETS
4. Let 2 be any finite or infinite system of forms, not all
zero. There exist ascending sets in 2; for instance, every
non-zero form of 2 is an ascending set. Among all ascending
sets in 2, there are, by the final remark of 3, certain ones
which have a least rank. Any such ascending set will be
called a basic set of 2.
The following method for constructing a basic set of 2
can actually be carried out when 2 is finite. Of the non-
zero forms in 2, let AI be one of least rank. If AI is of
class zero, it is a basic set for 2. Let AI be of class
greater than zero. If 2 contains no non-zero forms reduced
with respect to AI, then A l is a basic set. Suppose that
such reduced forms exist; they are all of higher class than AI.
Let -4 2 be one of them of least rank. If 2 has no non-zero
forms reduced with respect to AI and A 2 , then A l9 A 2 is
a basic set. If such reduced forms exist, let A$ be one of
them of least rank. Continuing, we arrive at a set (1) which
is a basic set for 2.
If Ai 9 in (1); is f c ^ ass greater than zero, a f orm F will
be said to be reduced with respect to the ascending set (1)
if F is reduced with respect to every Ai, i 1, - , r.
Let 2 be a system for which (1), with AI not of class
zero, is a basic set. Then no non-zero form of 2 can be
reduced with respect to (1). Suppose that such a form, F,
exists. Then F must be higher than AI, else F would be
an ascending set lower than (1). Similarly, F must be higher
than AS, else A i9 F would be an ascending set lower than (1).
Finally, F is higher than A r . Then AI, - , A r , F is an
ascending set lower than (1). This proves our statement.
I. IRREDUCIBLE SYSTEMS 7
Let -5" be as above. We see that if a non-zero form,
reduced with respect to (1), is adjoined to 2, the basic sets
of the resulting system are lower than (1).
Throughout our work, large Greek letters not used as
symbols of summation will denote systems of forms.
REDUCTION
5. In this section, we deal with an ascending set (1) with A
of class greater than 0.
If a form O is of class p>0, and of order m in y p , we
shall call the form dG/dy pm the separant of 0. The coef-
ficient of the highest power of y pm in G will be called the
initial of (?.*
The separant and initial of G are both lower than G.
In (1), let Si and It be respectively the separant and initial
of At, i=l, , r.
We shall prove the following result.
Let G be any form. There exist non-negative integers
Si) ti, i = 1, -, r, such that ivhen a suitable linear com-
bination of the Ai and of a certain number of their derivatives,
with forms for coefficients, is subtracted from
ft 3 * . . . S S r 7 f l ... T*r Q
1 r 1 r *
the remainder, R, is reduced with respect to (1).
We may limit ourselves to the case in which G is not
reduced with respect to (1).
Let Ai be of class pi, and of order m t - in y Pi , i = l,-,r.
Let j be the greatest value of i such that G is not reduced
with respect to Ai. Let G be of order h in y pj .
We suppose first that h>mj. If &i = h mj, then A^, the
/fith derivative of Aj, will be of order h in y pj . It will be
linear in y Pi h, with Sj for coefficient of y p ^. Using the al-
gorithm of division, we find a non-negative integer v such that
* Later we shall have occasion to use other symbols than y^ , , y n
for unknowns. If the unknowns in a problem are given listed in the order
u, v, , w, then w will play the role of y p , above, in the definitions of
separant and initial for a form effectively involving w.
2
8 ALGEBRAIC DIFFERENTIAL EQUATIONS
Sp O = C l A ( fJ + D l
where A is of order less than h in y pj . In order to have
a unique procedure, we take v\ as small as possible.
Suppose, for the moment, that pj<n. Let a be an
integer with Pj<a^n. We shall show that A is not of
higher rank than O in y a . We may limit ourselves to the
case in which A4~0. Also since Sj is free of y a , we
need treat only the case in which y a is actually present
in G. Let G be of order g in y a . Then the order of A
in y a cannot exceed g. If A were of greater degree than G
in y ag , Ci would have to involve y ag in the same degree
as A and Ci Af l> would contain terms involving y ag and
yp/i which could be balanced neither by A nor by S] 1 G.
This proves our statement.
If DI is of order greater than mj in y p , we find a relation
with A of lower order than A in y pj and not of higher
rank than A (or 6) in any y a with a>pj. For uniqueness,
we take v 2 as small as possible.
Continuing, we eventually reach a J5, t , of order not greater
than MJ in y p , such that, if
we have
(4) ^ G =
Furthermore, if a>pj, D u is not of higher rank than G in y a *
If D u is of order less than MJ in y p , D u is reduced with
respect to A/ (as well as any Ai with i>j). If A* is of
order w/ in y^, we find, with the algorithm of division,
a relation
1 D
u
with ^ reduced with respect to Aj, as well as -4/+i,
For uniqueness, we take tj as small as possible.
I. IRREDUCIBLE SYSTEMS
We now treat K as O was treated. For some ?<,/, there
are s t , t t such that Hffl^K exceeds, by a linear combination
of Ai and its derivatives, a form L which is reduced with
respect to AI, AI^I, , A r . Then
exceeds L by a linear combination of AI, Aj and their
derivatives.
Continuing, we reach a form R as described in the state-
ment of the lemma.
Our procedure determines a unique R. We call this R
the remainder of G with respect to the ascending set (1).
SOLUTIONS AND MANIFOLDS
6. Let Z represent any finite or infinite system. The
forms in 2 need not all be distinct from one another.*
When the forms of 2 are equated to zero, we obtain a
system of differential equations, which we shall represent
symbolically by 2 0.
In studying the totality of solutions of 2 = 0, it will be
of fundamental importance to have a sharp definition of
solution. Let y l , - - , y tl be functions, analytic throughout an
open region 93, whose points are in 51, which render each
form of 2 zero when substituted into the form. The entity
composed of 33 and of yi, , y n will be called a solution
of - = 0. Thus two systems y^, , y n which are identical
from the point of view of analytic continuation, will give
different solutions if they are not associated with the same
open region. For instance, if we take an open region 33 t ,
interior to 33, and use, throughout 93i, t/i, , j/n as defined
for 33, we get a second solution of 2 = O.t
* What we are really considering then, is a system of marks, each
mark being associated with a form. Two distinct marks may be associated
with identical forms.
t In Chapter VII, we shall, at one point, adopt a different definition,
calling any set of n formal power series, convergent or divergent, a solution,
if they yield zero when substituted formally into the forms of -T. Many
of our results hold for this definition.
10 ALGEBRAIC DIFFERENTIAL EQUATIONS
By a solution of 2, we shall mean a solution of 2 = 0.
The totality of solutions of 2 will be called the manifold
of 2 (or of 2 = 0).
If 2 t and 2 2 are systems such that every solution of 2 t
is a solution of 2 9 , we shall say that 2 9 holds 2\.*
COMPLETENESS OF INFINITE SYSTEMS
7. In 7 10, we prove the following lemma:
LEMMA. Every infinite system of forms in y lf , y n has
a finite subsystem whose manifold is identical with that of
the infinite system.^
An infinite system of forms whose' manifold is identical
with that of one of its finite subsystems will be called com-
plete.% Infinite systems which are not complete will be called
incomplete. In what follows, we assume the existence of in-
complete systems, and force a contradiction.
8. The system obtained by adjoining forms 6?i, , G m to
a system 2 will be denoted by 2 + Gi-\ \-G m .
We prove the following lemma:
LEMMA: Let 2 be an incomplete system. Let F : , , F s be
such that, by multiplying each form in 2 by some product of
non-negative powers of FI, , F s , a system A is obtained
which is complete. Then 2-\-FiF 2 F 8 is incomplete.
Let 2+Fi F s be complete, and let it hold and be
held by its finite subset
(5) F t ~.F 9 , Hi,-",JSi.
The presence of JF\ F 8 in (5) is legitimate, f or if 2 + FI - F 8
has the same manifold as a system r, it has the same manifold
as r+F l ... F s .
Let
(6) K 19 -, K v
* If Si has no solutions, every system will be said to hold Si .
t See 124 for a comparison of this lemma with a result of Tresse.
t If some finite subsystem has no solutions, the system will be considered
complete.
The product of powers of FI, , F, may, of course, be different
for different forms of S.
I. IRREDUCIBLE SYSTEMS H
be forms in 2 such that the forms of A which they yield,
after the above described multiplications, form a system <Z>
which is held by A. If some K* are not among the Hi in (5)
we may, as was seen above, adjoin them to the Hi. Similarly,
any Hi not present in (6) may be adjoined to (6). We shall
thus assume that (6) is identical with
(7) JZi, .--, H t .
Let L, in 2, not hold (7). Now some
Ff' ... FS*L
holds <Z>, and 0> holds (7). Then F^ - F S L holds (7).
Consequently certain solutions of FI F s which are so-
lutions of (7) are not solutions of L. Thus L does not hold
(5). This proves the lemma.
9. We prove the following lemma:
LEMMA. Let 2 and 2-\- F - F 3 both be incomplete. Then
at least one of the systems 2-{-Fi, - , 2-\-F s is incomplete.
We may evidently limit ourselves to the case of s = 2.
Let 2 + Ft and 2 + F 2 both be complete. Let <J> if i = 1, 2,
be a finite subset of 2 such that 2 + Fi holds <Z>< + .Fi.
Then 2 + jft holds (0>i + 0>i) + *i and 2 + F 2 holds
(0>i + 0> 8 ) + F 2 .* Now every solution of (0>! + <&*) + -Fi F 2
is a solution of (<Z>i+ <Z> 2 ) + FI or a solution of (<2>i-|-<Z> 2 ) + F 2 .
As 2 + F! jF 2 holds 5 + .Fi and ^ + F 2 , then 2 + F x F 2 holds
((Pi + <Z> 2 ) + F! F 2 . This proves the lemma.
10. Let us consider the totality of incomplete systems of
forms in y l9 -, y n . According to the final remark of 3,
there is one of them, 2, whose basic sets ( 4) are not
higher than those of any other incomplete system. Let (1)
be a basic set of 2. Then AI involves unknowns, else AI
would have no solutions, and 2 would be complete.
For every form of 2 not in (1), let a remainder with re-
spect to (1) be found as in 5. Let A be the system com-
posed of the forms of (1) and of the products of the forms
* (<D! -f (P,) consists of the forms present either in <J>i or in <0 S .
12 ALGEBRAIC DIFFERENTIAL EQUATIONS
of 2 not in (1) by the products Si 1 2 f r r used in their re-
duction. Let n be the system composed of (1) and of the
remainders of the forms of 2 not in (I).
Now -Q must be complete. If not, it would certainly have
non-zero forms not in (1). Since such forms would be re-
duced with respect to (1), then (1) could not be a basic set
of & ( 4). This means that & would have ascending sets,
hence basic sets, lower than (1) and 2 would not be an in-
complete system with lowest basic sets.
If H is a form of A not in (1), and R the corresponding
form in &, then H and E have the same solutions in
common with (1). This means that A and -Q have the same
manifold and also that A is complete.
The lemmas of 8, 9 show us now that either some
2 + Si is incomplete or some 2 + It is incomplete. But, for
every i, Si and Ii are distinct from zero, and reduced with
respect to (1). Then, by 4, the basic sets of 2 + Si and
of 2-\-It are of lower rank than (1). This proves the funda-
mental lemma stated in 7.
NON-EXISTENCE OF A HILBERT THEOREM
ii. One might conjecture, on the basis of Hilbert's theorem
relative to the existence of finite bases for infinite systems
of polynomials,* that, in every infinite system 2, there is a
finite system such that every form of 2 is a linear combi-
nation of the forms of the finite system, and their derivatives,
with forms for coefficients. We shall show that this is
not so.
We consider forms in a single unknown y. (See first foot-
note in 2.)
Consider the system
We shall show that no form of this system with n>l is
linearly expressible in terms of the forms which precede it,
and their derivatives.
* van der Waerden, Moderne Algebra, vol. 2, p. 23.
I. IRREDUCIBLE SYSTEMS 13
We notice that all of the forms, and all of their derivatives,
are homogeneous polynomials of the second degree in the yt.
Also, if the weight of yiyj is defined as i+j'j the pth derivative
of yiyj will be isobaric, with its terms of weight i -\-j-\- p.
Now if
the terms in the At, JBi, etc., which are not independent of
the yi may be cast out, for they produce terms of degree
greater than 2. Again, considering the weights of the various
forms, we find that
(pn-2 ft*
(8) yy*+i = Ci 2n-2 (yi y*) H ----- 1" C^-i -2 (yn-iy w ) ,
with Ci which are independent of the yj. Now the (2n 2)d
derivative of y\y* contains a term y\yi n , and none of the
other derivatives in (8) yields such a term. We conclude
that Ci = 0. Continuing, we find every Ci to be zero. This
proves our statement.
IRREDUCIBLE SYSTEMS
12. A system 2 will be said to be reducible if there exist
two forms, O and H such that neither O nor H holds 2 but
that G H holds 2. Systems which are not reducible will be
called irreducible. The system of equations 2 = 0, and also
the manifold of 2, will be said to be reducible or irreducible
according as 2 is reducible or irreducible.
Example 1. Let 2 9 in the unknown y, consist oiy\ y
and i/ 2 2. (See Introduction, p. v.) Let OH hold 2.
Let Gj and J3i be the remainders for O and H respectively
with respect to j/ 2 2. Then 6?i and HI will be at most
of order 1 and GiH L holds 2. Then as every y = (# a) 8
with a constant is a solution of 2, GiHi, if not zero, must
be of order 1. Let K be the remainder of O^ HI with respect
to y\ 4y. One can prove now without difficulty that K
vanishes identically. Then GiHi is algebraically divisible
14 ALGEBRAIC DIFFERENTIAL EQUATIONS
by y\ 4y. As y\ 4y is algebraically irreducible,* one of
GI, HI must be divisible by y\ 4?/. This means, since the
initial and separant of y% 2 are both unity, that one of
G, H is a linear combination of the two forms of 2 and their
derivatives. Then 2 is irreducible in every field.
Example 2. We use two unknowns, u and y. Let 2 =
be uy u\ = 0. Differentiating, we find
Uiy -\~uyi 2ieiW 2 = 0.
Multiplying the last equation through by y and using 2 = 0,
we have
= 0.
Certainly ^ does not hold 5. Neither does
since it vanishes only for ?/ = 0, if u = 0. Thus 2 is reducible
in the field of rational constants. We call attention to the
fact that u y u\ is algebraically irreducible, and of order in y.
THE FUNDAMENTAL THEOREM
13- A system 2 will be said to be equivalent to the set of
systems 2 i , , 2 S if 2 holds every ^ and every solution
of 2 is a solution of some 2 it Thus, two systems with the
same manifold are equivalent to each other.
We prove the following fundamental theorem.
THEOREM. Every system of forms is equivalent to a finite
of irreducible systems.
Let the theorem be false for some 2. Then 2 is reducible.
Let GI and 6? 2 be such that G^G*, but neither GI nor G 2 ,
holds 2. Then 2 is equivalent to the set
(9)
Thus at least one of the systems (9) is reducible. A re-
ducible system in (9) will be called a system of the first class.
* That is, irreducible as a polynomial in y\,y.
1. IRREDUCIBLE SYSTEMS 15
There must be a system of the first class, which, when treated
like 5, yields one or two reducible systems obtained by
adjoining two forms to 2. The reducible systems obtained
through two adjunctions, we call systems of the second class.
Some of the systems of the second class, when treated like 2,
must yield reducible systems obtained from ^ by three ad-
junctions, that is, systems of the third class. We proceed
in this manner, forming systems of all classes.
There must be a system of the first class whose forms are
contained in systems of all classes higher than the first. Let
2+fli, where HI is either Q or (? 2 , be such a system of
the first class. One of the systems of the second class which
contains the forms of 2 + HI must have its forms contained
in systems of all classes higher than the second. Let
2+-Hi+-H"s be such a system. Let an H p be found, in this
way, for every p . Then the system */*, composed of
2, -Hi, H 2 , , Hp,
is incomplete. For, if *P held
with a finite subsystem of 2 and i < < i q , then
would hold
(10)
This cannot be, since Hi+i does not hold (10). This proves
our theorem. One will notice that the proof involves making
an infinite number of selections.*
UNIQUENESS OF DECOMPOSITION
14. Let a system 2 be equivalent to the set of irreducible
systems
(11) 2i,---,2,.
We may suppose, suppressing certain of the 2i if necessary,
that no 2i holds a Sj with j^i. We shall then call each 2i
* See 70.
16 ALGEBRAIC DIFFERENTIAL EQUATIONS
an essential irreducible system held by 2, and we shall call (11)
a decomposition of 2 into essential irreducible systems.
We shall prove that the decomposition (11) of 2 into essential
irreducible systems is essentially unique. That is, if '/2 l? ,$
is a second decomposition of 2 into essential irreducible systems,
then t = s and every &t is equivalent to some 2j.
We shall show that there is some &* which holds 2 l9 If
there were not, then each t - would have a form which
would not hold ^\ . Such forms being selected, their product
would hold each *, consequently 2, thus 2 t . This is im-
possible if 2i is irreducible and none of the forms holds 2 l .
Then let 1^ hold 2 lu Now 12 U similarly, must be held
by some 2 t , which must be 2 l9 since no 2 t with i^l
holds 2^. Thus 2 t und &i are equivalent. The uniqueness
is proved.
EXAMPLES
15. We shall consider some examples involving one un-
known, y, in which, in spite of the fact that the systems
decomposed consist of a single form, the results are not un-
instructive.
Example 1. Let 2 = be y\ y = 0. By differentiation,
we find, for any solution of -5",
22/22/3 2/1 = 0,
(12) 2y 2 jfc + 2^ y a =0,
(13) 27/22/5 + 62/32/4 2/3 = 0.
Multiplying (13) by 2y 3 and substituting into the result the
expression for y\ found from (12), we find that
2/2(42/3^5 12yJ + 8y 4 1) = 0-
Thus 2 is equivalent to the set of two systems
y\y,
As the only solution of 2 l is y 0, ^ is irreducible in
every field. We shall see in the next chapter that the
I. IRREDUCIBLE SYSTEMS 17
manifold of 2$, which is the "general solution" of y\ ?/,
is irreducible in every field. We note that 2* does not
hold 2^.
Example 2. Let 2 == be y\y* y = 0. We find,
with a single differentiation, that ~ is equivalent to the two
systems :
- yy*y, ?/i 2/3 .
2i and Z, are irreducible in any field (as above). We call
attention to the fact that the form in - is linear in y$.
Example 3. The form y\(yi y) decomposes into the
essential irreducible systems y l and T/ L - y. These two
systems have the solution y = in common.
The above examples might lead one to conjecture, that
any 2 can be decomposed into irreducible systems by means
of differentiation and elimination. We shall see in Chapter VII
that this is actually so.
RELATIVE REDUGIBILITY
1 6. Let A be any system of forms. A system 2 will be
said to be reducible relatively to A if there exist forms G
and H in A such that GH, but neither O nor H, holds
2. Otherwise 2 will be said to be irreducible relatively
to A.
For instance, if 2 is the form (dy/dx)* 4y, 2 is redu-
cible in the field of rational constants if A is the set of all
forms in y of orders 0, 1, 2, but is irreducible in any field
if A is the set of all forms of orders 0, 1. (See example 1,
12.)
We see, as in 13, that every system is equivalent to
a finite number of systems irreducible relatively to A.
The decomposition into relatively irreducible systems need
not be unique. For instance if A is the form 1, the system
in the above example, which is relatively irreducible, is equi-
valent to the two relatively irreducible systems y\ 4y, y l
and y\ 4y, y 2 2.
18 ALGEBRAIC DIFFERENTIAL EQUATIONS
If 2 consists of forms in^/, 2 can be resolved into relatively
irreducible systems whose forms belong to A. If A is such
that the product of two forms of A belongs to A, such a
decomposition is essentially unique in the sense of 14.
Wherever the contrary is not stated, we shall deal with
irreducibility as defined in 12. That is A will consist of
all forms with coefficients in &.
ADJUNCTION OF NEW UNKNOWNS
17. One might ask how the theory of a system 2 in the
unknowns y l9 , y n is affected if new unknowns vi, , n
are introduced, and -5" is regarded as a system of forms in
the yi, vi. For instance, will the decomposition (11) of 2
into irreducible systems, when the yi are the unknowns,
continue to be such a decomposition when the unknowns are
the yi and Vi?
To show that the answer to this question is affirmative,
we consider an irreducible system 2 of forms in the yi and
prove that it remains irreducible when the unknowns are
the yij Vi. We represent 2, considered as a system in the
yij v^ by 2'.
Suppose that G and H are forms in the yi, Vi such that
neither holds 2', but that OH holds 2'. Let and H be
arranged as polynomials in the Vij, with coefficients which
are forms in the yi.
We note that the solutions of 2' are obtained by adjoining,
to every solution yi, , y n of 2, arbitrarily assigned func-
tions t>i, , vt.
Evidently, then, the terms of G and H in which the
coefficients hold 2 can be suppressed and the modified G
and H will be such that neither holds 2', while GH does.
We assume thus that no coefficient in O or H holds 2.
As 2 is irreducible, it will have a solution for which no
coefficient in G or H vanishes. Then we can certainly replace
the Vijj in O and H, by rational constants, so as to get two
forms, (?i and HI, in the y^ neither of which holds -2". On
I. IRREDUCIBLE SYSTEMS 19
the other hand, since we can construct analytic functions n
for which the vy in OH have any assigned values, at any
given point, and since OH holds 2', it is necessary that
(rtjffi hold 2. This proves that 2 f is irreducible.
FIELDS OF CONSTANTS
18. In later work, it will at times be desirable to assume
that cV contains at least one function which is not a constant.
We establish now a result which will permit us to make
this assumption with no real loss of generality.
Suppose that e? consists purely of constants. Let ^ be
the field obtained by adjoining x to r, that is, the totality
of rational functions of x with coefficients in &. We shall
prove that if a system 2 of forms in r is irreducible in r,
then 2 is irreducible in S L .
We start by proving that if G, of the type
(14)
with the Bt forms in F, holds 2, then each Bi holds 2. Let
(15) yi(x),'-> yn(x)
be any solution of 2. Since the forms in 2 have constant
coefficients,
where c is a small constant, will also be a solution of
This means that, for any solution (15),
where is any constant, vanishes identically in x. Then
each Bi must vanish identically in x. This proves our
statement.
* We shall not encumber our discussions with references to the areas
in which the solutions are analytic.
20 ALGEBRAIC DIFFERENTIAL EQUATIONS
Now, let G and H be forms in &i such that GH holds 2.
We have to prove that one of G, H holds 2. We may
evidently limit ourselves to the case in which G is given
by (14) and H by
with the Ci forms in S r .
Suppose that neither G nor JT holds 2. In G and JFf,
let every BI and ft which holds 2 be suppressed. For the
modified G and //, GH will still hold 2. Then
Since neither B m nor G"? holds 2, B m C s cannot hold 2, so that
GH cannot hold ^'. This proves that ~ is irreducible in ^ .
CHAPTER II
GENERAL SOLUTIONS AND RESOLVENTS
GENERAL SOLUTION OF A DIFFERENTIAL EQUATION
19. We consider a form A in t/ t , , y n of class n,
which is algebraically irreducible in r, that is, not the product
of two forms, each of class greater than 0, and each with
coefficients in r.
We are going to introduce the notion of the general
solution of A.
We write y n = y and, if w>l, we write q = n 1,
yt = MI, i = 1, , q.
Our definition of the general solution will appear, at first,
to depend on the order in which the unknowns happen to
be arranged,* at least, on the manner in which y is selected
from among the unknowns effectively present in A. But it
will turn out, finally, that the object which we define is
actually independent of such order.
20. Let 8 and / be, respectively, the separant and initial
of A .t A solution of A for which neither S nor / vanishes
will be called a regular solution of A.
We shall make plain that regular solutions of A exist.
Let A be of order s in z/. Since SI is of lower degree than
A in y s , 81 and A, considered as polynomials in the un-
knowns and their derivatives, are relatively prime. ThenJ
there is a B^ which, if s>0, is of order less than s in y
and which, if s = 0, is free of y, such that
* That is on the manner in which the subscripts 1 , , n are attributed
to the unknowns.
t See footnote in 5.
JBocher, Algebra, p. 213; Perron, Algebra, vol. 1, p. 204.
21
22 ALGEBRAIC DIFFERENTIAL EQUATIONS
(1) B = C(SI) +
We shall use the symbol to designate values of x at
which all coefficients of the forms in (1) are analytic, and
the symbol fa] to represent any set of numerical values which
one may choose to attribute to the unknowns and their
derivatives present in A, omitting y s . Let ?, fy] be taken
so that JjS^O. We can then find a number such that
A = for y s = , when the other symbols in A are replaced
by their values fa]. Then, by (1), $/ cannot vanish for
In particular, since /S^O, we see by the implicit function
theorem that there exists a function
(2) y s = f(x\ MI, -, ys-d,
analytic for the neighborhood of ?[^] and equal to at ? [*/],
which makes A = for the neighborhood of ?fy].
Let functions i, , u q , analytic at J, be constructed
which have for themselves, and for their derivatives present
in A, at ?, the corresponding values in [y]. Let (2) be
considered as a differential equation for y, and let y, , f/ 5 _i
be given, at , the values which correspond to them in [//].
Then, by the existence theorem for differential equations,
(2) determines y as a function analytic at ?, and the func-
tions HI, -, Uq\ y will constitute a regular solution of A.
21. Let Q and if be such that every regular solution of A
is a solution of OH. We shall prove that either every regular
solution of A is a solution of or every regular solution of A
is a solution of H.
Let (?i and HI be, respectively, the remainders of G and H
with respect to A. Then, as some S p I t G exceeds G^ by
a linear combination of A and its derivatives,* every regular
solution of A which annuls GI annuls G] similarly for HI
and H.
If, then, we can show that either GI or HI is identically
zero, our result will be proved. Suppose that neither GI
* At times we shall, without explicit statement, use symbols, as p
and t above, to represent appropriate non-negative integers.
II. GENERAL SOLUTIONS AND RESOLVENTS 23
nor HI vanishes identically. As G l and HI are of lower
degree than A my sj G^R^IS, as a polynomial, is relatively
prime to A. Hence, we have
B =
with S^O and free of y s . As in the discussion of (1), we
can build a solution of A for which G H^ IS does not vanish.
But GiHi, like GH, vanishes for every regular solution
of A. This contradiction proves our result.
22. It follows immediately, from 21, that the system of
all forms which vanish for all regular solutions of A is
irreducible. A belongs to this system. The irreducible
manifold composed of the solutions of this system will be
called the general solution of A = (or of A).
We show that every solution of A for which S does not
vanish belongs to the general solution.
Let B be any form which vanishes for all regular solutions.
Then some S l B exceeds, by a linear combination of derivatives
of Ay a C of order at most s in y. C vanishes for all regular
solutions of A. We have
(3) IPC = DA + E,
with E reduced with respect to A. Since E vanishes for
all regular solutions of A, E, by the discussion of (1), must
vanish identically. Thus, as / cannot be divisible by A,
C is so divisible. This means that S 1 B holds A, so that
B vanishes for every solution of A with 8 ^ 0. This proves
our statement.
As we shall see later, the general solution may contain
solutions with $ = 0.
Let ^i be the system of all forms which vanish for all
solutions of A with 8^0. In a decomposition of the
system A, /Sinto essential irreducible systems, let 2 2 , , 2 t be
those systems which are not held by 2 lt Then
(4) 2 lf 2*, -.., 2 t
is a decomposition of A into essential irreducible systems.
24 ALGEBRAIC DIFFERENTIAL EQUATIONS
Thus, the general solution of A is not contained in any
other irreducible manifold of solutions of A. In a decompo-
sition of A into essential irreducible systems, those irreducible
systems whose manifolds are not the general solution are held
by the separant of A.
We shall prove that the general solution of A is independent
of the order in ivhich the unknowns in A are taken.
Suppose that, m being some unknown other than y effectively
present in A, we order the unknowns so that Ui conies last.
With this arrangement, let the manifold of 2j in (4) be the
general solution of A, and let S' be the separant of A.
Suppose that j ^ 1 . Then S' holds 2^ , while 8 holds
2 2 , ..., 2 t . Thus SS f holds A. As was seen in the dis-
cussion of (1), this cannot be, since neither 8 nor S' is
divisible by A. This proves our statement.
In Chapter VI, we shall secure a characterization of the
solutions of A with 8 = which belong to the general
solution. For the present, we limit ourselves to the statement
that any solution of A towards which a sequence of solutions
with 8 4 1 converges uniformly in some area, belongs to the
general solution. In short, any form which vanishes for all
solutions with 8 ^ will vanish for the given solution.
We can now see that, in the examples in 15, the systems 2 2
are irreducible. In each case, the separant vanishes only
for y = Q, and y = gives no solution of 2 2 . Thus, in
each case, the manifold of 2 9 is the general solution.
CLOSED SYSTEMS
33. A system 2 will be said to be closed if every form
which holds 2 is contained in 2.* Given any system <Z>,
the system 2 of all forms which hold <Z> is closed, and has
the same manifold as <P. Hence no generality will be lost,
in the study of manifolds, if we deal only with closed systems.
The only closed system devoid of solutions is the totality
of forms with coefficients in S r .
*A given form is supposed here to occur only once in 2.
II. GENERAL SOLUTIONS AND RESOLVENTS 25
A system which contains non-zero forms, and possesses
solutions, will be called non-trivial.
Let 2 be a non-trivial closed system in y t , -, y n Let
(5) AI, AS, - - -, A r
be a basic set of 2. Then AI is of class greater than 0.
A solution of any ascending set which does not cause the
separant or initial of any form of the set to vanish, will be
called a regular solution of the ascending set.
We shall prove that every regular solution of (5) is a solution
of 2.
Let Si and It be respectively the separant and initial of At.
Let G be any form of 2. Then the remainder of G with
respect to (5) is a form of 2. This remainder, reduced with
respect to (5), must be zero ( 4). That is, some S^ 1 I* r r G
is a linear combination of the Ai and their derivatives. Then
G vanishes for every regular solution of (5). Q. E. D.
Suppose now that 2 is irreducible. As no Si or I, holds 2,
the product of the Si and 7* does not hold 2. It follows,
that, if 2 is irreducible, (5) has regular solutions.*
Furthermore, if 2 is irreducible, any form which vanishes
for all regular solutions of (5) belongs to 2. For, if G is such
a form, Si I r G holds 2 so that G holds 2.
Thus, if 2 is irreducible, then 2 is the only closed irreducible
system for which (5) is a basic set.
ARBITRARY UNKNOWNS
24. Let 2 be a non-trivial closed system in y lf , y n *
There may be some y, say yj, such that no non-zero form
of 2 involves only yj ; that is, every form in which yj appears
effectively also involves effectively some yu with k^j. If
there exist such unknowns yj, let us pick one of them,
arbitrarily, and call it U L .
There may be a y, distinct from u v , such that no non-zero
* In Chapter VI, we determine which solutions of (5) other than the
regular ones are solutions of 2.
26 ALGEBRAIC DIFFERENTIAL EQUATIONS
form of 2 involves only u and the new y. If there exist
such ijj let us pick one of them, arbitrarily, and call it u*.
Continuing, we find a set MI, -, u q (q<ri), such that no
non-zero form of -5" involves the m alone and such that given
any unknown */;, not among the M*, there is a non-zero form
*of 2 in yj and the HI alone.
Let the unknowns distinct from the te$, taken in any order,
be represented now by y i9 - , y p , (p-\-q = n).*
We now list the unknownst in the order
(6) u^'-tUq] yi, -',y p .
We shall speak generally as if 11* exist. It will be easy
to see, in every case, what slight changes of language are
necessary when there are no m.
Of the non-zero forms in 2 involving only y and the w,
let A l be one of least rank. There certainly exist forms of 2
of class q + 2 which are reduced with respect to y t ; for
instance any non-zero form in y* and the iti alone is of this
type. Of such forms, let A 2 be one of least rank.
Continuing, we build a basic set of 2,
(7) A i9 A t , ,A P .
We shall say that Ai introduces yi.
We shall call t/i , , w^ a set of arbitrary unknowns.
THE RESOLVENT
25. In this section, we assume that F does not consist
purely of constants.
Let 2 be a non-trivial closed system. Let the unknowns
be ui, , u q \ j/i, -, y pj with the m arbitrary unknowns.
We are going to show the existence in $ of functions
(8) f*i, , (*p
* It will be seen in 30 that when 2 is irreducible, q does not depend
on the particular manner in which the w< may be selected,
t See remarks on notation, 2 and 5.
II. GENERAL SOLUTIONS AND RESOLVENTS 27
and the existence of a non-zero form G, free of the yi, such
that either
(a) There exist no two solutions with the same area of analyt-
icily, and with the same Ui,
^i> ' > *V 2/i> ' > y'pj
w/ // ti
MI, , u q \ ?/i, - -, yp
for which G does not vanish and in which, for some i,
y'i is not identical with y'i , or
(b) such pairs of solutions exist, and for each pair,
(10) /ii (y[ y'i) + + t*p (y'p y'p)
is not zero*
We consider the system of forms obtained from 2 by
replacing each yi by a new unknown z*. We take the
system & composed of the forms of 2, the forms in the zi
just described, and also the form
in which the A t are unknowns. That is, & involves 3p-\-q
unknowns, namely the Ui, yi, Zi, it.
Let A be any closed essential irreducible system which 12
holds. Suppose that one of the forms yi Zi, i 1, -,p,
does not hold ^. We shall prove that A contains a non-
zero form which involves no unknowns other than the m
and A*-
If A contains a form in the m alone, we have our result.
Suppose that A contains no such form.
Since A has all forms in 2, A has, for j = 1, - -,p,
a non-zero form Bj in yj and the m alone. Let Bj be taken
so as to be of as low a rank as possible in yj. Then Sj,
the separant of J5/, is not in A.
Similarly let C/, j = 1, , p, be a non-zero form of A
in gj and the m alone, of as low a rank as possible in gj.
Letting gj follow the Ui in C/, we see that the separant 8j
of Cj is not in A.
* If no Ui exist, this is to mean that if 2 has a pair of distinct solutions,
(10) does not vanish for the pair. We take 6r = 1 in this case.
28 ALGEBRAIC DIFFERENTIAL EQUATIONS
To fix our ideas, suppose that yi z is not in A. Con-
sider any solution of A for which
(which is not in A} does not vanish. For such a solution,
we have
From (11) we find, the ^'th derivative of A u an expression
(12) A i; = g;(A 9 , ., Aj,; y n -, y,,; z l7 -, z p ),
in which QJ is rational in the A t -, t/ t -, ^ and their derivatives,
with coefficients in &. The denominator in each QJ is a power
of 2/1 ^i-
Let each * be of order n in y< and each d be of order
Si in *f.
If a QJ involves a derivative of yi of order higher than n,
we can get rid of that derivative by using its expression in
the derivatives of yi of order n or less, found from Bi = 0.
Similarly, we transform each QJ so as to be of order not
exceeding Si in z*, i = 1, -, p.
The new expression for each QJ, which will involve the m,
will have a denominator which is a product of powers of
y\ *i; Si, Sj, i 1, ,!>. Let # be the maximum of the
integers n, Si. Let
h
Let k be the total number of letters y</, Zij which appear in
the relations (12), transformed as indicated. Then h>k.
We consider the first h of the relations (12).* (That is,
we let j = 0, 1, -, h 1). Let D, an appropriate product
of powers of y t * u the Si, Si', be a common denominator
for the second members of these relations. We write
<*> * = f '
*When / = 0, (12) is (11).
II. GENERAL SOLUTIONS AND RESOLVENTS 29
j = o, , & 1. Let D and the Ej, be written as poly-
nomials in the A: letters y^, zy present in them, with coefficients
which are forms in A 2 , , h p and the w,. Let m be the
maximum of the degrees of these polynomials (total degrees
in the y#, zy).
Let a represent a positive integer to be fixed later. The
number of distinct power products of degree ma or less, in
k letters, is*
(U) (ma + k).~(ma+l) ^
Using (13), let us form expressions for all power products
of the Ay in (13) of degree or less. Let each expression
be written in the form
(15) Jr-
Then F, as a polynomial in the y^, zy, will be of degree
at most ma.
The number of power products of the h letters A v -, of
degree a or less, is
fia\ \ a i */ * * * \ a i I/
(16) .
Now (14) is a polynomial of degree k in , whereas (16) is
of degree h in . As h>k and as w, A, A: are fixed, (16)
will exceed (14) if is large. Let a be taken large enough
for this to be realized.
If now the F in (15) are considered as linear expressions
in the power products in the yy, zy, we will have more
linear expressions than power products. Hence the linear
expressions F are linearly dependent. That is, some linear
combination of the F, with coefficients which are forms in
Aj lp and the ut, not all zero, vanishes identically.
The same linear combination of the power products of the
Jiy will vanish for the solution of A for which (11) was
written. Now, this last linear combination is a form H
* Perron, Algebra, yoL 1, p. 46.
30 ALGEBRAIC DIFFERENTIAL EQUATIONS
in the m and ^ which is not identically zero, since the power
products of the 1*\j in H are distinct from one another.
Thus
is in A, so that H is in A. This proves our statement.
Let AI , , A r be a decomposition of into closed
essential irreducible systems. Let A\ , , A s each not con-
tain some form yi Zi and let ^5+1, , A r each contain
every y t Zi. Let Hi be a non-zero form in A iy i = 1, . . ., s,
involving only the m and Aj. Let K Hi - H s .
We wish to show the existence in S of p functions
Mi, % Mp such that, when each A$ is replaced by ^ in K,
then JST does not vanish identically in the H{.
Let K be written as a polynomial in the t^-, with forms
in the ^ as coefficients. Let L be one of the coefficients
in K. If we can fix each ^ in ? so that L does not vanish,
our result will be established.
Let f be any non-constant function in $, and let a be
a point of 51 at which is analytic and has a non-vanishing
derivative. Given a sufficiently small circle with a as center,
any function y, analytic in the circle, can be expressed as
a power series in with constant coefficients. Then <p can
be approximated uniformly within the circle by a polynomial
in . Thus if m is a sufficiently large integer, and if
ko, , km, i 1, -, p are arbitrary constants, L cannot
vanish identically in the ty if each A* is replaced in L by
Otherwise L would vanish if the ^ are any functions analytic
in the above circle. Thus there must be integral values of
the tij for which L does not vanish. Every polynomial in
with integral coefficients is in r . This shows the existence
of the required /^.
The solutions of for A/ = ^;, j = 1, - , p, will be the
solutions of the Ai for */ = f*/. Now, the solutions with
A/ = pj of ^i, - - , v/ 5 have u* which cause to vanish the
II. GENERAL SOLUTIONS AND RESOLVENTS 31
form obtained by putting A; = /*; in K. The solutions
of 4 8 +i, , A r , even with A/ = ^, have yi ^, i = 1 , , p.
When every Aj contains every yt Zi, we take (7=1,
/*i = = Pp = 0.
We have thus the result stated at the head of this section.*
26. We shall now relinquish the condition that & contain
a non-constant function
Let us assume that m exist. We are going to prove the
existence of forms 0, M\, - - , M p , in the in alone, with G^Q,
such that, for two distinct solutions (9) for which O does not
vanish,
(17) Mi(y[-y' l ')+ - - +M p (y' p y'A
is not zero.
The discussion of 25 holds through the construction of
the form K. We are going to prove the existence of forms
Mi, - , M p in the m alone, such that, when A, is replaced
by Mi in K, the resulting form 6 is not identically zero.
Let K be arranged as a polynomial in the A^-, with forms
in the ui, A 2 , , h p as coefficients. Let U\H be the highest
derivative of u^ which appears in any of the coefficients.
Let k be an integer greater than h. Then, if AI is replaced
by MM, K becomes a form K in the in and A 2 , ..., i p ,
which is not identically zero. Similarly, if we replace A 2 in
KI by a sufficiently high derivative of MI , we obtain a non-
zero form K 2 in the t^ and A 8 , , A p . Replacing^, , ^>
in succession by sufficiently high derivatives of HI, we ob-
tain a non-zero form O.
Continuing as in 25, we see that the solutions of n in
which A/ = Mj, j = 1, -, p are the solutions of the Ai
which satisfy A; = Mj. Now the solutions with A; = Ji}
of -^, -, A 9 have m which cause (7 to vanish. The so-
* The following example shows that 2 may have many solutions with
given Ui, and that a G may exist, such that, for #4=0, there is only
one solution for given t*<. Let the unknowns he Wi, w s , y\. Let 2 con-
sist of all forms which hold MI y\ U* . Let G = HI. Then w t , w is
a set of arbitrary unknowns. If Wi = Wj = 0, y\ may he taken arbi-
trarily, but, for given M I? u s with (74=0, there is only one y\*
32 ALGEBRAIC DIFFERENTIAL EQUATIONS
lutions of ^5+1, - , ^rj even with A/ = Mj 9 have yi = Zi,
i = 1 , - - , p . This proves our statement.
27. The results of 25, 26 permit us to state that if
either
(a) o? does not consist purely of constants or
(b) there exist w,-,
then triads of forms G, P, Q, exist with O and P not in
2, and # free of the y, such that, for any two distinct
solutions of 2, with the same w, such that neither G nor
P vanishes, the expression Q/P yields two distinct functions
of x. For instance, if (a) holds, we can take P= 1 and
It will essentially increase the generality of our work to
use general forms P. The following is a non-trivial example
in which P is of class greater than 0. Let r be the to-
tality of rational functions of x. Let the unknowns be y 9
y 2 and let 2 consist of all forms which hold y u und y%.
The solutions are y l = c, y* = d, with c and d constant
but arbitrary. We take = 1. If
the expression Q/P gives distinct functions of x for distinct
solutions of 2 with P^ 0.
In certain cases in which & consists purely of constants
and in which no m exist, there may exist no pair P, Q as
described above. For instance, let & be the totality of con-
stants. Let the unknowns and 2 be as in the preceding
example. The yy are all zero for j > for every solution.
We therefore lose no generality in seeking a P and Q of
order zero in y l9 y*. For any such P and Q, Q/P will yield
the same result, for infinitely many distinct pairs of con-
stants t/i, t/ 2 .
In developing the theory of an irreducible system 2 for
the case in which & has only constants and there are no
Ui y two courses are open to us. If we adjoin x to r, then,
II. GENERAL SOLUTIONS AND RESOLVENTS 33
by 18, 2 will remain irreducible in the enlarged field.
Working in the enlarged field, we can secure a P and Q.
Again, by 17, we can introduce a new unknown MI and 2
will remain an irreducible system. After either type of ad-
junction, the theory which follows will apply.
28. In 28, 29 we deal with a non-trivial closed irre-
ducible system 2. We assume that either
(a) r does not consist entirely of constants, or
(b) arbitrary unknowns exist.
We take a triad 6, P, Q, as in 27.
We introduce a new unknown, w, and consider the system A
obtained by adjoining the form Ptu Q to 2. Let be
the system of all forms in w, the u i and y { which vanish
for all solutions of A with P^O.* We shall prove that
12 is irreducible.
Let B and C be such that BC holds 12. For s appropriate,
P S B minus a linear combination of Pw Q and its derivatives,
is a form R free of w. We obtain similarly, from a P*C,
an 8, free of w. Then US vanishes for every solution of 2
with P^O, since every such solution yields a solution of 12.
Hence PES holds 2, so that either R or 8 is in 2. If
R is in 2, P 8 B holds 12. Hence B vanishes for all solutions
of A with P^O, so that B is in 12. Thus 12 is irreducible.
We notice that those forms of 12 which are free of w
are precisely the forms of 2. In particular, 12 contains no
non-zero form in the m alone.
We are going to show that 12 contains a non-zero form
in w and the m alone.
Let Bi, i 1, -, PJ be a non-zero form of 2 involving
only y { \ u v , u , of minimum rank in y r Let S i be the
separant of Bi.
Consider any solution of 12 for which PS L Sp does not
vanish. For such a solution, we have
* Of course, forms in & may also vanish when P 0.
34 ALGEBRAIC DIFFERENTIAL EQUATIONS
For the jth derivative of w, we have an expression
(18) J =
Using the relations Bi = 0, we free each Qj from derivatives
of each y i of order higher than the maximum of the orders
of Qj P and B i in y r Each w. will then be expressed as
a quotient of two forms, the denominator being a product
of powers of P, Si, - , S p . If we use a sufficient number
of the relations (18), as just transformed, we will have
more w, than there are y. in the second members. Using
the process of elimination employed in 25, we obtain a non-
zero form K in w\ Mi,-,w tf which vanishes for every
solution of & with PSi S p ^ 0. As P/Si S p is not
in , and as & is irreducible, .XT is in .
29. We now list the unknowns in in the order
and take a basic set for Q,
(19) A, 4,...,^.
Here, w, j/ 1? , y p are introduced in succession. (See final
remarks of 24.)
If A is not algebraically irreducible, we can evidently
replace it by some one of its irreducible factors. We assume,
therefore, that A is algebraically irreducible.
We are going to prove that AI , - , A p are of order
in y l9 - , y p , and, indeed, that A. is of the first degree in y r
Thus, since A i with i>\ will be of lower degree in yj than
Aj with j < i y each equation A. = expresses y i rationally
in terms of iv\ Wi, , u q and their derivatives.
The determination of the manifold of 2 will in this way
be made to depend on the determination of the general
solution of A = 0, which equation will be called a resolvent
of 2.*
* If A is any system equivalent to 2, we also call A = a resolvent
of A.
II. GENERAL SOLUTIONS AND RESOLVENTS 35
Suppose that A t is of order higher than zero in y lm
Consider any regular solution of (19) for which PG does
not vanish. By the final remarks of 23, such regular
solutions exist. Let 2jf be any point at which the functions
in this solution and the coefficients in
P, 0, A, A,, ..-, A p
are analytic, and for which, if 8 and / are the separant
and initial of A, Si and It those of Ai,
POSS, . .. Split "- Ip + 0.
Without changing w or the m in the solution, we can alter
slightly the values at ? of the derivatives of y , in A^ , other
than the highest, and obtain a second regular solution of
(19) with PG 0. That is, we can solve A = for ^
with the modified initial conditions, substitute the resulting
yi into A 2 , solve A 2 = with the same initial conditions
for 2/2 which obtained in the first regular solution,* and,
continuing, determine each yt. This is nothing but an
application of the implicit function theorem, and of the
existence theorem for differential equations. Thus, we would
have two distinct solutions of , with the same ut, with
PG 4"- 0, and with the same w. This contradicts the
fundamental property of the triad G, P, Q.
Hence, A is of order zero in t/i. Similarly, every At is
of order zero in yi. Furthermore, as At is of lower rank
in yj than Aj for j<i, each At is of zero order in yj for
3 ^ *'
We shall now prove that each Ai is linear in yi.
We start with A p . Suppose that A p is not linear in y p .
Let PI be the remainder for P with respect to (19). Then
every regular solution of (19) which causes either of the
forms P, PI to vanish, causes the other to vanish.
* That is, with the same values at for y^ and all its derivatives but
the highest.
36 ALGEBRAIC DIFFERENTIAL EQUATIONS
If we can show that the ascending set
(20) A, A 19 , Ap-i
has a regular solution
1, , Uq] W\ 7/i, -, y p -i
for which A p has two distinct solutions in y p with S p I p Pi ^ 0,
we shall have forced a contradiction.
If we cannot get two distinct solutions of this type, it
must be that for every regular solution of (20) with I p ^ 0,
the equation A p = has a solution in y p for which S p PI G
vanishes.*
Let C be a remainder for S p PI G with respect to A p con-
sidered as an ascending set. Then C is of zero order in
every yi and is of lower degree than A p in y p . Every
common solution of S p PI G and A p is a solution of C. We
note that C, like S p , PI and A p , is not of higher order in
w than A.
Of all forms not in fi, of zero order in the yi, which are
of lower degree than A p in y pj not of higher order in w
than A, and which, for every regular solution of (20) with
IP ^ have an annulling function y p in common with A p ,
let D be one which has a minimum degree in y p . Then D
must be at least of the first degree in y p , else I P D would
vanish for every regular solution of (19) and would be in .
Let K be the initial of D. Then K is not in , else D
would not be of a minimum degree in y p . For m appro-
priate,
K m A p = ED + F,
with E of lower degree than A p in y p and F of lower degree
than D in y p . Every annulling function y p of A p and D
makes F vanish. Then F must be in .
Thus ED must be in fl, so that E, which is not zero,
is in . Let
(21) E = H Q + H iy +--
* Because (19) has regular solutions, (20) has regular solutions with
1,4=0.
II. GENERAL SOLUTIONS AND RESOLVENTS 37
with the Hi forms free of y p and of order in w not greater
than that of A. We understand that Ht ^ 0.
As K m A p is of higher degree in y p than F, the initial of
ED is identical with that of K m A p and hence is not in .
Then Ht is not in .
Evidently a non-negative integer a t exists such that, when
a suitable multiple of A p _ l is subtracted from I^_ 1 H i9
i = Q, -,t, the remainder is of lower degree than A p -i
in 2/^-1.* In the same way, we find integers a 2 , , a p -\;
a such that when a suitable linear combination of the forms
of (20) is subtracted from
i = y . . . , t, the remainder is reduced with respect to (20). As
..! IMS
is in , we see that contains a form
with each Hi reduced with respect to (20) and with Hi not
in (hence not 0). As t is less than the degree of A p in
y p , EI, which is not zero, is reduced with respect to (19).
This contradiction ( 4), proves that A p is linear in y p .
We now consider ^lp-i, assuming that it is not linear in
y p -i. Since PI is of lower degree in y p than A p , PI is free
of y p . It must be that, for every regular solution of
(22) A,A
19
with Ip-i ^ 0, Ap-i = has a solution in yp-\ for which
/Sp-i I p PiG vanishes. The proof continues as for A p .
In dealing with A p -i, we consider that both PI and I p
are free of y p -i. The proof continues as above.
Thus every Ai is linear in yi , and each yi has an repression
rational in w\ u i9 , u q and their derivatives, with coeffi-
cients in &.
* Note that Jp-i is free of y p .
38 ALGEBRAIC DIFFERENTIAL EQUATIONS
We notice that, if
u l7 -, u q \ iv\ */i, , y p
is a solution of &, then u i9 -, u q \ w belongs to the general
solution of A.*
For, if a form K in w and the 114 vanishes for every
solution in the general solution of A, then K vanishes for
every regular solution of (19), and so is in Q.
When Q is in 2, w = is a resolvent. Then each yi
is rational in the w#.
The introduction of the resolvent accomplishes the
following:
(a) It reduces the study of an irreducible system to the study
of the general solution of a single equation. Of course
for solutions of 2 with P = 0, there may be no cor-
responding w and, for other solutions of 2, the initial
of some At may vanish. We shall gain information as
to these exceptional solutions in Chapter VI.
(b) It leads to a theoretical process for constructing all irre-
ducible systems ( 32).
(c) It creates an analogy between y l9 , y p and a system
of p algebraic functions of q variables. It is well known,
in short, that, given such a system of algebraic functions,
we can find a single algebraic function in terms of which,
and of the variables, the functions of the system can be
expressed rationally.
(d) It furnishes an instrument useful in the solution of for-
mal problems.
INVARIANCE OF THE INTEGER q
30. We consider a non-trivial closed irreducible system 2
in any field r .
We propose to show that, if arbitrary unknowns exist,
the number q of arbitrary unknowns does not depend on the
manner in which the in are selected.
* Here we consider A as a form in to and the u< alone.
II. GENERAL SOLUTIONS AND RESOLVENTS 39
Let Ui exist. It will suffice to prove that, given any
unknowns among the ui and yi,
there exists a non-zero form in 2 which involves only the Zi.
We form a resolvent for 2. As m exist, this is possible.
Let us consider the regular solutions of (19). Every zi in
such a solution has a rational expression in tii, , u q \ w.
If a Zi happens to be a H, say M/, the expression for that
Zi is simply iij. We write
(23) Zi = Qi (;; U L , - , u q ), (i = 1, , g+1)-
On differentiating (23) repeatedly, we get expressions for the
Zi) which are rational in the wj and mj. Making use of the
relation A = 0, we transform these relations so as not to
contain derivatives of w of order higher than r, where r is
the order of A in w.
None of the expressions thus obtained will have a deno-
minator which vanishes for a regular solution of (19).
Since there are q~\- 1 of the z,-, and only q of the Mi, it
follows that if we differentiate (23) often enough (and then
transform), the Zij will become more numerous than the ug
and w, MI, - -, w r *
It follows, as in 25, that there exists a non-zero form
in the Zi which vanishes for all regular solutions of (19).
The form thus obtained belongs to 2. The invariance of q
is proved.
The assumption that 2 is irreducible is essential. For in-
stance, consider the system
MI 2/i = ^ 2 2/2 = u 3 2/2 = 0,
in the unknowns MI, M 2 , M 3 ; yi, y%. These equations impose
no relations either upon the yi or upon the m. Thus MI,
M2, MS and 2/1, 2/a are two sets f arbitrary unknowns.
4
40 ALGEBRAIC DIFFERENTIAL EQUATIONS
ORDER OF THE RESOLVENT
31. We work with any non- trivial closed irreducible system 2
for which triads G, P, Q, and therefore resolvents, exist.
Considering 2 as a system in the m and t/i, let
(24) ^i,---,4,
be a basic set for 2, the separant and initial of Ai being Si
and Ii respectively.
Let the order of Ai in yi be n. Let
We shall prove that every resolvent of 2 is of order h in w.
We begin by proving that n contains a non-zero form
in w\ ?(i, -, u q whose order in w does not exceed h.
Consider any solution of for which
(25) PS, . Spli Ip + 0.
For such a solution, we have
(26) iv = |.
We propose to show the existence of forms R and T each
of order not exceeding r< in y<, i = 1, , p, such that, for
any solution of for which (25) holds, T is not zero and
(27) w = f .
Let Qi and PI be the remainders of Q and P respectively,
relative to (24). Let Qi be obtained by subtracting a linear
combination of the Ai and their derivatives from
and let P t be obtained similarly from
r p p p.
II. GENERAL SOLUTIONS AND RESOLVENTS 41
Then, if (25) holds, we have
a s? 1 i Tp
(28) w = -^~~ i *-.
For H and T in (27) we take the numerator and denominator
in (28) respectively.
We find, from (27), for the jth derivative of w, an ex-
pression
(29) wj =
If Uj is the remainder of Bj with respect to (24), we can
write (29)
(30) wj = %r,
where Wj is a product of powers of T, Si - - I p .
Consider (27) and the first h relations (30). Let D be
a common denominator for the second members in these
h -\~ 1 relations. We write
(3D wj = JL,
3 = 0, , h.
Let D, the Ej and the Ai be written as polynomials in
the yij with coefficients which are forms in the ut. Let m
be the maximum of the degrees of these polynomials.
For convenience, we represent the nth derivat"'* of yi by *<
Let Ai be of degree vt in z\.
Let be a positive integer, to be fixed later. In (31),
let us form all power products in the wj of degree a or less.
Let the expression for each power product be written in
the form
(32) -.
Then each F is a polynomial in the yy, of degree not ex-
ceeding met.
Let each expression (32) be written
(33)
4*
42 ALGEBRAIC DIFFERENTIAL EQUATIONS
Consider a particular F, and let it be written as a poly-
nomial in z p . Suppose that its degree d in g p is not less
than m. Then, as A p = for the solution of Q which we
are considering, we have, letting
the relation
(34)
with the Ji free of z p , we may write the numerator in (33)
in the form
(35) (J I p + ... + J d I p e*)I?>-i.
Since I p is of degree less than m in the y</, each term in
the parenthesis in (35) is of degree less than m(a-\-l).
We replace J d l p z* by J d Mz d ~ v * in (35). As J d is of
degree not exceeding ma d in the yu and as M is of
degree at most m, then J d Mz d ~ Vp is of degree less than
w(+l) in the yy. Thus, (33) goes over into
where FI is of degree less than w( + l) in the y^ and of
degree less than d in z p . If the degree of JP\ in ^ is not
less than m, we repeat the above operation. After t^
operations, we get an expression
TT jmnt
^
""
with Zf of degree less than m in z p and of degree less than
w( + <) in the y</. The numerator in (36) is of degree in
the y$ less than
w(4-0 + w(m < 2m 2 .
Thus, if we let A = Z> Jj 1 , we can write each power
product in the Wj, of degree or less, in the form
< 37) -
II. GENERAL SOLUTIONS AND RESOLVENTS 43
where JTis of degree less than 2w 2 in the y {j and of degree
less than m in z p .
We now write each expression (37) in the form
(38)
'
T)a
J
Tim a
and employ, with respect to Zp-\, the procedure used above.
We find for each expression (38), an equivalent expression
with D t = J>i l 2 ^ and with L of degree less than 4m 8 a in
the y {j and of degree less than m in ^ and z p __ r Continuing,
we find an expression for each power product of the wj
W
(40) -^r
where W is of degree less than 2 p m p+1 a in the y^ and of
degree less than w in #, i = 1 , , p. Let c represent 2^ m p+1 .
The number of power products in t , ,, of degree
less than m in each letter, is m p .
Hence the number of power products of the y.. of degree ca
or less, and of degree less than m in each ^, is not more
than
(ca-\-h) ... (ca + 1)
(41)
hl
This is because the ly .. with ^' < r t . are h in number. On the
other hand, the number of power products of degree a or
less in the h + 1 Wj is
(42) /7 , -,\,
As (42) is of degree h + l in and (41) only of degree h y
(42) will exceed (41) for a large. This, as we know from
25, implies the existence of a non-zero form of in w
and the m alone, of order not exceeding h in w.
44 ALGEBRAIC DIFFERENTIAL EQUATIONS
This shows that the order in iv of the resolvent A =
does not exceed h. Suppose that the order of A is k < h.
For each y v we have an expression
(43) y t = f
with d and Di forms in w\ MI, , i^, of order not ex-
ceeding it in w. We obtain from (43) expressions for the y^.,
j = 0, , n- 1, which are rational in the wj,mi, with
powers of the A as denominators. Using the relation 4 = 0,
we depress the orders in w of the numerators until they do
not exceed fc. The transformed expressions will have de-
nominators which are power products of the A and 8.
By an elimination we obtain a non-zero form W in the
2/i, HI which belongs to ., hence to 2. This W, which
is of order less than r t - in each y,-, is reduced with respect
to (24). This is impossible.
We have thus proved that the order in w of every resolvent
is h.
We say now that, when UL, - , u q are selected, the quantity
f*i + + Tp does not depend on the manner in which the
subscripts 1, ,# are assigned to the remaining unknowns.
This follows immediately from what precedes, except that
we have to prove that, when no m exist and of has only
constants, n-f -\-r p is independent of the order of they*.
What we do is to introduce a new unknown, u lt 2 will
remain irreducible, iii will be an arbitrary set, and (24) will
remain a basic set. The methods above then apply.
The degree of the resolvent in WH does depend on P and Q.
Consider, for instance, the system, irreducible in the field of
all rational functions
As the manifold is ^ = x + a, y* = (x + a) 2 , we may evi-
dently take w = yi. The resolvent becomes w l 1 = 0.
On the other hand, if we take w = 2/1 + 2/2 , the resolvent
becomes of the second degree in w\.
II. GENERAL SOLUTIONS AND RESOLVENTS 45
The order of the resolvent depends on the choice of the ?*.
For instance
2/112/2 =
is irreducible in the field of all constants. If we let HI = y% ,
we get a resolvent of the first order. If we let u = y L ,
we get a resolvent of zero order.
CONSTRUCTION OF IRREDUCIBLE SYSTEMS
32. We shall establish a result which is, to some extent,
a converse of the result of 29.
Let A be an algebraically irreducible form in MI, , u q ;
w, effectively involving w. Let
(44) yi : = --, i = 1, ,#,
where the Pi and Q are forms in MI, , u q \ w and where Q
does not vanish for every solution in the general solution
of A. Let MI, ---lUg] w be any solution in the general
solution of A which does not annul Q. For this solution,
we obtain, from (44), functions j/i, , y p .
Let be the system of forms in the MJ, y<, w which vanish
for all HI, yi, w. We shall prove that .,is irreducible.
Let GH hold . If we substitute (44) into (?, we get
with T a form in MI, , M ff ; w and Z7 a power of Q. We
have, similarly, H = V/W. For any lit, w, T V vanishes.
Then TVQ vanishes for every solution in the general solution
of A. Thus either T vanishes for every such solution of A,
or V does. Consequently one of G, H must vanish for all
w, y^ w. Hence & is irreducible.
By the method of 25, we can show that contains
non-zero forms in the M,-, y< alone. The system 2 composed
of all such forms and the zero form is a closed irreducible
system. What is more, the theory of resolvents shows that
46 ALGEBRAIC DIFFERENTIAL EQUATIONS
every closed irreducible system in yi, , y n can be obtained
in this way. We have thus a theoretical process for con-
structing all closed irreducible systems.
IRREDUCIBILITY AND THE OPEN REGION 21
33- The question might be raised as to whether 2, irre-
ducible in for the open region 21, can be reducible in &
for some open region 2li in 21. We shall show that the
answer is negative.
Without loss of generality, we assume 2 non-trivial and
closed. Also, we assume, adjoining x to & if necessary, that
oF does not consist purely of constants.
Let a form K vanish for all solutions of 2 which are
analytic in a part of 2li. We shall prove that K vanishes
for all solutions of 2. Suppose that K is not in 2. We
construct a resolvent, and consider (19). Let Jfi be the re-
mainder of K with respect to (19). Then K v is not divisible
by A. On the other hand,
(45) K, 8 II, I p
(as in 29), vanishes for every solution of A Avhich is ana-
lytic in a part of 2Ii. This is impossible, because (45) is
not divisible by A ( 20).
Thus if P and Q are forms such that PQ vanishes for all
solutions of 2 analytic in a part of 2li, then PQ vanishes
for all solutions of 2. This means that either P or Q is in
2, so that 2 is irreducible in 2d.
CHAPTER III
FIRST APPLICATIONS OF THE GENERAL THEORY
RESULTANTS OF DIFFERENTIAL FORMS
34. In algebra, in developing the theory of resultants of
systems of polynomials, it is necessary to deal with poly-
nomials whose coefficients are indeterminates. So, in con-
nection with resultants of pairs of differential forms, we
shall find it desirable to deal with general forms. For our
purposes, it will be convenient to define a general form in
y as one of the type
A = OQ + MI y + #2 P^ H \- a n P n ,
with n 2> 1 where the ai are indeterminates and where the
Pi, i = 0, , n, (Po = 1, PI = y), are distinct power
products in y and its derivatives. By an indeterminate, we
mean a symbol which can be replaced, when it is desired,
by an arbitrarily assigned analytic function.*
Consider a second general form
B = Jb +
We propose to find a condition upon the a t - and bi, ne-
cessary for the existence of a common solution in y of A
and B.
Let the ca and bi be considered now as unknowns, and let
A and B be considered as forms in the a*, bi and y in any
field \ We shall show that the system
* It is understood that, when the a* are replaced by, analytic functions,
the replacing functions have a common domain of analyticity. In defining
a general form, we do not use the notion of a coefficient field.
47
48 ALGEBRAIC DIFFERENTIAL EQUATIONS
(1) A, B
is irreducible.
Let X and T be forms such that X Y holds (1). We have
(2) fl = i y a n P n \ b = hy b m Q m .
If of and & are replaced in X and in Y by the second
members in (2), there result two forms Xi and FI in
(3) i, , a n \ &i, , & w ; ^
such that Xi YI holds (1). But as the unknowns (3) may
be taken arbitrarily, as analytic functions, and a , b be
determined by (2) so as to make A = B = 0, it must be
that Xi YI vanishes identically. Then one of X i9 F t must
vanish identically, and one of Z, Y must hold (1).
Let 2 be the system of all forms which hold (1). Using
(2), it can be shown, by the method of 25, that 2 contains
non-zero forms in the 04 and bi alone.
On the other hand, 2 contains no non-zero form in
(4) i , ; , a n ; b Q , , b m .
Suppose that such a form, C, exists. Let the unknowns in
(4) be taken as analytic functions, with &i , , b m not all
zero, so that (7^0. Then certainly J? = has a solution
in y. Using any such solution ?/, we can determine 0$ so
that ^4 = 0. C cannot exist.
Suppose now that $ is the field of all rational constants.
Let (4) be the arbitrary unknowns, and let
(5) R, U
be a basic set for 2, R and U introducing a and y
respectively.
We assume that R is algebraically irreducible and that
its coefficients are relatively prime integers. This determines
R uniquely, except for algebraic sign. For, let 8 be any
other form which satisfies the conditions placed on R. Ther
8 and R are of the same rank. The remainder of S with
III. APPLICATIONS 49
respect to B, being in 2, must be zero. Then J being the
initial of R, some I p S is divisible by R. Hence 8 is
divisible by R, and as the coefficients in 8 are relatively
prime, we have 8 R. We suppose the sign of R to
be fixed according to any suitable convention, and treat R
as unique.
We shall call R the resultant of A and B.
We shall now prove that U is of order in y and, indeed,
that U is linear in y.
Let c?i be the field obtained by adjoining x to 3\ We
form a resolvent for (1) in Si, using a w defined by
(6) w = ao + py,
with ^ a rational function of x. Let the resolvent be V =
and let y = N/M, with M and JV forms in the a*, 6$, and te;.
The system
(7) A, B, iv a Q py
is equivalent to the system
(8)
A solution of the first two forms in (8) will satisfy F= 0,
My ^=0. If, then, F A , Jtfi, -ZVi are the forms which
result from V, M, N respectively on replacing w by a and
0i by OI + P,
(9) V^M.y N,
will be a basic set in ^ for 5 X , the totality of all forms
in TI which hold (1).
Evidently "Pi cannot be of higher rank in a than R. This
implies that J/i and NI are of lower rank in OQ than JB.
Let M^y NI be written in the form
with the & and T, forms in the ca and 6i, with integral
coefficients and with a a polynomial in x.
50 ALGEBRAIC DIFFERENTIAL EQUATIONS
The numerator in (10) holds 2. By 18, each SiyTi
holds 2. Let j be such that Sj^O. Then SjyTj is
a non-zero form of 2 reduced with respect to J2. This proves
that U, in (5), is linear in y.
Thus, for A and B to have a common solution in y, it is
necessary that
OQ, , dn\ VQ, ' ' ' 9 Om
be a solution in the general solution of the resultant of A and B.
If (to, - - , b m is such a solution, and if it does not annul
a certain fixed form in a , , &m,* then A and B have
a single solution in common, which can be expressed rationally
in terms of a , , b m , with integral coefficients.^
We prove now that the resultant of A and B is a linear
combination of A, B and a certain number of their derivatives,
the coefficients in the linear combination being forms with
integral coefficients.
In R, let a and b Q be replaced respectively by
A aiy -a n Pn, B 6 t y b m Qm,
and let R be expanded as a polynomial in A, B and their
derivatives. The term not involving A, B, or their derivatives,
will be a form in the unknowns (3) which holds (1). As we
saw, such a form vanishes identically. This gives our result.!
The methods of Chapter V permit the actual construction
of resultants.
ANALOGUE OF AN ALGEBRAIC THEOREM OF KRONECKER
35- It is a theorem of Kronecker that, given any system
of algebraic equations in n unknowns, there exists an
equivalent system containing n + 1 or fewer equations. We
present an analogous theorem for differential equations.
* The coefficient of y in U.
t We are using the expression "single solution" in the sense of analytic
function theory rather than in the sense of 6.
JFor a theory of resultants of linear differential forms, see Heffter,
Journal ftir die r. u, a. Mathematik, vol. 116 (1896), p. 157.
KOnig, Algebraische Grqften, p. 234.
III. APPLICATIONS 51
THEOREM. Let $ contain a non-constant function. Let 2
be any system of forms in y l9 , y n . Then there exists
a sijstem Q>, composed of n-\- I or fewer forms, ivhose manifold
is identical with that of 2. If 2 consists of a finite number
of non-zero forms
(ii) *\, ..-,*;,
then a system Q) exists which is composed of I\ and n or fewer
linear combinations, ivith coefficients in 3r, of F 2 , - , F r .
We shall need the following lemma, which applies to
a perfectly general field.
LEMMA. Let W be a closed irreducible system, mui,--',u q \
yi> ' ' > ypj with u j ' > u q a se t f arbitrary unknowns.
Then there exists a basic set for W
in ivhich, if an AI involves a HJ effectively, the partial deri-
vative of Ai, with respect to the highest derivative of Uj in At,
does not belong to *P.
We show first how to choose AI . From among all forms
of ^ of class # + 1, we select those of least rank in y la
Prom the forms just selected, we choose such as have a least
rank in u qj and continue, taking the ranks in M^-I, , MI,
in succession, as low as possible. For A L , we take any of
the forms thus obtained. Obviously AI fulfills our require-
ments. In choosing A 2 we first take all forms of V of class
# + 2 which are reduced with respect to AI. From these,
we select such as have a least rank in y* and continue as
above with respect to the m. We find thus an A 2 as specified.
In the same way, we determine A*, , A p , in succession,
to meet the requirements of the lemma.
36. Returning to the proof of our theorem, we limit our-
selves, as, according to 7, we may, to the consideration of
the finite system (11). Introducing r 1 new unknowns,
^2 9 > v r , we consider the system 2 1 , composed of the two
forms
(12) F
52 ALGEBRAIC DIFFERENTIAL EQUATIONS
Let ,&! be used to represent the system (11) when the un-
knowns are the y*, v*.
Let -5*1 be resolved into closed essential irreducible systems
>/!, -, A 9 . Suppose that 1 does not hold some A, say .///.
We say that, given any n 1 unknowns among y ly , y n ,
then .//; contains non-zero forms in those n 1 unknowns
and the Vi. For instance, suppose that./// does not contain
a non-zero form in
(13) 2/1, ' ', */n-r, Vt,'-,V r .
Then (13) will be a set of arbitrary unknowns for .///, so
that 4j will have a basic set consisting of one form, B,
which introduces y n . We take B algebraically irreducible.
Then the general solution of B is the manifold of .///.
We shall prove that B does not involve the Vi. For
instance let B involve ty.
According to 22, the general solution of B is the same
manifold for all arrangements of the unknowns. Thus far
we have treated the unknowns as if y n followed (13). Let
us now give them the order
Consider any regular solution of B. If we vary the yi in
this solution, and any finite number of their derivatives
arbitrarily, but slightly, at some point ?, we can, using the
v t , , v r -i of the given regular solution, determine v r so
as to get a second regular solution of B.* But this con-
tradicts the fact that J\ holds Jj. Thus B is free of the v^
This mfeans that, given any solution y l9 , y n in the
general solution of B considered as a form in the yi alone,
and given any analytic functions t; 2 , , v r , the given yt f
Vi constitute a solution of ^//, hence a solution of
t;. *! + + tv*V.
* Of course we have to construct new analytic functions yi which
assume, with their derivatives, the modified values at {.
III. APPLICATIONS 53
But, as the vi can be given arbitrarily, F 2 , , F r must
vanish separately for the given y*. This means that H
holds AJ % Our statement is proved.
Let A l9 , A q be those A t which are not held by &i.
Consider any n 1 of the y,-,
We extract from each A^ i = 1, , #, a non-zero form
in the yi of (14) and the Vi. Let the 3 forms thus obtained
be multiplied together. We obtain thus, for every set (14),
a form which vanishes for every solution of 2 l which is not
a solution of Q l . Now, as of contains non-constant functions,
we can so fix the Vi in of that every form obtained above
becomes a non-zero form in its set (14). Let the system of
forms thus obtained, from the various sets (14), be denoted
by <V
Let /7i represent the system of two forms in the t/ alone
which 2} becomes when the vi are fixed definitely as above.
Then every solution of #t which is not a solution of 2 is
a solution of (D lt
The unknowns Vi, whose role was episodic, now disappear
from our discussion. We examine n L . We introduce r 1
new unknowns w*, , w r and consider the system 2 2 obtained
by adjoining to Hi, the form
Let 2 be used to represent 2, considered as a system in
the yt, wi. Let 2 2 be decomposed into closed essential
irreducible systems A^ , J 8 . Suppose that 2 does not
hold some A, say Aj. We say that, given any n 2 of
the yi 9 then Aj contains a non-zero form in those n 2 y<
and the wi. Imagine, for instance, that Aj does not contain
a non-zero form in y l9 -, y n -z and the wt.
Every form of O^ is in Aj. For, let Q be any form of <Z>t
and let FU be any form of 2 which does not hold Aj.
Consider any solution of Aj. If the yi in the solution annul
jRk, they annul F k . If the yi do not annul F k , then, since
54 ALGEBRAIC DIFFERENTIAL EQUATIONS
they annul each form of II l9 they must, as seen above,
annul G. Thus GFk holds A^ so that G is in Aj.
Thus Aj has a form in any n 1 of the unknowns yi.
Hence y t , , y n -z and the wt are a set of arbitrary un-
knowns for yfjj and Aj has a basic set B l9 J9 2 , which in-
troduces y n -i and y n respectively. Let B 1 and B 2 be taken,
as in the lemma of 35, so that, if one of them involves
a Wi, its derivative with respect to the highest derivative
of that wi is not in Aj. In addition, let J? t be algebraically
irreducible.
We say that BI and B^ are free of the Wi. For instance,
suppose that B^ involves Wk effectively. Let G be the form
of <Z>i in 2/1, , j/n-i. Then G vanishes for every solution
in the general solution of B lu * This, cannot be, for, ordering
the unknowns in BI so that wu comes last, we find that
the j/i , - , y n -i , in any regular solution of B , can, together
with any finite number of their derivatives, be given slight,
but otherwise arbitrary, variations, at some point , and
WH then be determined for a second regular solution of BI.
Again, suppose that B 2 involves Wk* Let S be the
derivative of B 2 with respect to the highest derivative of
Wk in B* .t Consider a regular solution of BI , B 2 for which
8 does not vanish. Let if be the form of i in y l9 , y n -2 9
y n alone. Let be a point for which the functions in the
solution and the coefficients of BI , B 2 , H are analytic, with
the coefficients of H not all zero, and for which neither 8
nor the separants and initials of BI, B 2 vanish. We can
modify yi, -, y n -<i slightly, but arbitrarily at , and deter-
mine t/n-i so as to get a new regular solution of BI. We
can then use the modified y lf - - - , y n -\ and, varying y n and
any finite number of its derivatives slightly, but arbitrarily,
at , determine Wk from B 2 = 0, securing another regular
solution of BI, jB 8 . Thus we can get a regular solution
of BI, Bt which does not annul H.
*The remainder of G with respect to BI holds Aj and thus is 0.
t Notice that S is not the separant of B t . We are using the unknowns
in their original order.
III. APPLICATIONS 55
Thus J5i and B 2 are free of the WL Then the y l9 , y n
in a regular solution of B i7 B 2 , with arbitrary analytic
functions w 2 , -, w r , give a solution of 2 2 . This means
that .F 2 , , F r all hold Aj 9 so that since FI, as a form
of 5 2 , holds ^//, 2 holds ^//. This contradiction proves
that Aj has a form in any n 2 of the y,-, and the Wi.
Let ^ , . - , v/ 7 be those systems A{ which are not held
by 2 . Consider any n 2 of the y<,
We extract from each A^ i=l 9 , q a non-zero form
in the yt of (15), and multiply together the q forms thus
obtained. We get, for every set (15), a form which is
annulled by every solution of 2 2 which is not a solution
of 2 . We fix the w in 3? so that each of the foregoing
forms becomes a non-zero form in its unknowns (15).
Let <Z> 2 be the set of forms thus obtained. Let // 2 be
the system which 2 2 becomes when the Wi are fixed as above.
Then every solution of ZT 2 which is not a solution of 2 is
a solution of cf> 2 .
We form a system 2 3 , adjoining to // 2 the form
where the Zi are unknowns. We introduce s in the expected
way. Let Aj be a closed essential irreducible system held
by 2> 8 which ^ 8 does not hold. We have to show that,
given any n 3 of the y^ there is a non-zero form in Aj
in those yi and the zi alone. Suppose that Aj does not
contain a non-zero form in yi, , y n -s and the #. Then,
as every form of <Z> 2 holds Aj, Aj has a basic set B ly B%, B$
which introduce y n -2, yn-\, yn in succession. Let this basic
set be selected as in the lemma of 35. Furthermore,
merely to abbreviate the proof, let us assume that BI is
algebraically irreducible. We see at once that B^ involves
no Zi. Also, if jB 2 involved a z/, we could practice arbitrary
slight variations on the yi, , y n _ 8 , y n -\ and their derivatives
5
56 ALGEBRAIC DIFFERENTIAL EQUATIONS
in a regular solution of Si , J? 2 and get a second such regular
solution. This cannot be, since every such regular solution
would have to be a solution of the form of <2> a in yi, , y w -8,
y/i-i. Finally, if S B involved a Zi> we could take yi, , y w -8,
y rt , and any finite number of derivatives, quite arbitrarily,
at some point, and get a regular solution of Si , J5 a , J5 8 .
This contradicts the fact that <Z> 2 has a form in the above yi
alone.
Thus, the z\ being properly fixed, we get two systems,
J7 8 and <Z> 3 , the latter containing a non-zero form in every
n 3 of the y,-,.such that II 3 holds 2 and that every solution
of 7/3 which is not a solution of 2 is a solution of <Z> 3 .
In /Tg, there are four forms.
Continuing, we find a system equivalent to 2, containing
at most n-\-l forms.
That w + 1 equations may actually be necessary, in con-
nection with n unknowns, is seen on considering the system
in y,
y? 4y, 2/ 2 2 >
which defines the general solution of yj 4y. If a single
form, G, had the manifold of this system, G would have to
be of the first order in y. Then G would have to be divi-
sible by y\ 4y, and so would admit the solution y = 0,
which does not satisfy the given system.
37 The assumption above that $ does not consist entirely
of constants is essential. For instance the system in y,
vu -y*> y i
has no solutions. Still for any pair of constants di and d 8 ,
the form
diy' + d 2 (y 1)
has solutions in common with y^.
However, the following result, which can doubtless be im-
proved, holds for fields of constants.
THEOREM. Let A be an algebraically irreducible form in
a single unknown y, the order of A in y being r. Then
III. APPLICATIONS 57
there exists a system of forms, consisting of A and of at
most r other forms, whose manifold is the general solution of A.
We shall need the following lemma:
LEMMA: Let 2 i9 , 2 8 be closed irreducible systems, none
of which holds any other, and let 2 be a closed system which
holds no 2i. Then there exists in 2 a form which holds no 2i.
We proceed by induction. The lemma is true for s = 1 .
We shall prove that the truth for s 1 implies the truth
for s. The truth for s 1 implies that each 2 i9 i = 1 , ., s,
has a form AI which holds no 2j with j \ i. Let Bi,
i = 1, -, s, be a form of 2 which does not hold 2*. Let
JTi A.\ AI i A.i-^\ AS, 'I L y , S .
Consider the form
C^ '-.: ~P /? I -I ~P 7?
which belongs to 2. Since PijBi does not hold 2 l9 and since
Pi for i > 1 holds -5"i , C does not hold 2^ . Similarly, C holds
no 2i.
Let A be resolved into closed essential irreducible systems,
2 9 2 19 9 2 a ,
2 having the general solution of A for manifold. Let Bi,
i = 1, , s be a non-zero form of lowest rank in 2i 9 so
that the manifold of 2i is the general solution of Bi. (We
may and shall assume that each Bi is algebraically irreducible.)
Each 2i is held by the separant S of A, hence by the resultant
(as in algebra) with respect to y r of A and S considered as
polynomials in y r . This means that each J?< is of order less
than r in y.
Let AI be any form of 2 which holds no 2t. We shall
examine the system A 9 A 1 . This system is equivalent to the
set of systems
Let d be the remainder of AI with respect to Bi. Because
A l does not hold 2 t , d$Q. As d holds 2i + A l9 the
resultant of Bi and Ci with respect to the highest derivative
58 ALGEBRAIC DIFFERENTIAL EQUATIONS
in Bi holds 2 { -f A l . Because Bi is algebraically irreducible
and of higher rank than Ci, this resultant is not zero.
This means that if A, A^ is resolved into closed essential
irreducible systems,
2 9 2 *9 ' ' '> 2 't>
each 2'i will contain a non-zero form of order less than r 1.*
Choosing now a form A 2 in 2 which holds no 2' it we form
the system A, A l9 A 2 and operate as above. Continuing, we
find that, after adjoining, to ,^4., r or fewer forms of 2, we
get a system of forms whose manifold is that of 2.
FORM QUOTIENTS
38. An expression A/B, where A and B are forms in
2/i, - , y n , with B not identically zero, will be called a/orm
quotient. Two form quotients will be considered equal if
they are equal as rational functions of the y</. It is easy
to see that, for A/B and CID to be equal, it is necessary
and sufficient that they yield the same analytic function for
given analytic y l9 -, y n which do not annul BD.
Let
(16) y = ^,
where A and B are forms in a single unknown u. The
question which we shall study is that of attributing a meaning
to y in the case in which u is such that both A and B
vanish.
The totality 2 of forms in y und u which vanish for all
solutions of
(17) By-A =
with B $ is an irreducible system. The manifold of 2 is
the general solution of the equation obtained on dividing (17)
by the highest commoji factor of A and B considered as
polynomials in the int.
*If r = 1, this means that there are no S.
f The results of Chapter VI will show that the manifold involved is
independent of the field employed.
III. APPLICATIONS 59
A function y will be said to correspond to a function u
through (16) if M, y is a solution of 2.
Example 1. Let
Then every analytic y corresponds to u when u = 0. For,
let be any analytic function, not identically zero. If k is
a non-zero constant, y = a^la when u = ka. Allowing k
to approach zero, we find that 0, i/ belongs to the
manifold of -2.* By taking suitably, we can make i/
become any desired analytic function (in some area).
Example 2. Let
v = .
y u
Referring to Example 2, 12, we see that, since UL does not
vanish for every solution in the general solution of uy u\,
then */ 2 + ?*!*/! 2w 2 y must. Thus, f or u = in the general
solution, we must have y = 0. If u = k, y approaches zero
uniformly as k approaches zero. Thus y = 0, and no other
function, corresponds to u = 0.
Example 3. Let
We find that no y corresponds to u = 0.
Example 4. Let
(ttJ + U,) (!+)
y =
y
We find, putting y 1 z, that
(1 8) U Ui Z = U\ (!+ W) + U\ .
Differentiating, we have
(19) u\e + uu l e l = u 2 P,
where
(20) P = 2 w 8 (ui+
* See next to last paragraph of 22.
60 ALGEBRAIC DIFFERENTIAL EQUATIONS
Multiplying (19) by uz, and using (18), we find
(21) ulg + uulg 1 = u 2 Q
where
(22) Q = u z P u 2 (u, z + u gj (M! + u).
We multiply (19) by HI and subtract from (21). Then
i(Q MiP) = 0.
Then Q UiP holds 2.
Suppose that u is a constant it distinct from 0. Then
Q u l P=Q implies Q==0. By (22), zP=0, so that,
by (20), uz 2 = 0. Thus * = and y = 1. If we let
M = k-\-hx, y approaches 1 as h approaches 0. Thus y = I
and no other function, corresponds to u when u is a non-
zero constant.
On the other hand, if we let u = A, we find that y ap-
proaches (!+)/ as h approaches 0. Thus every analytic
function y corresponds to u = 0.
39. As the general solution of an equation is completely
defined by a finite number of algebraic differential equations
we see that, if u makes A and B vanish, then either every
analytic y corresponds to u, or else, the functions y which corre-
spond to u are the totality of solutions of a system of alge-
braic differential equations, in the coefficients of which,
u figures.
We are going to study the circumstances under which no
y corresponds to a u which annuls A and B.
Writing A = A(u), B = J3(u), we let
(23) C(v) = A(u + v), D(v) = B&+V).
Then C and D are forms in v, with analytic coefficients
which are not necessarily in r . Also, v = makes C and
D vanish. Let
C= E+H, D = F+K,
where E and F contain respectively those terms of C and D
which are of lowest total degree in v and its derivatives.
III. APPLICATIONS 61
Suppose first that the degree of E is at least that of F.
Let be an analytic function which does not annul F when
substituted for v. Let v be replaced in C and D by ha,
where h is a constant. As h approaches 0, C/D will ap-
proach uniformly to an analytic function, which corresponds
to u through (16).
If E is of lower degree than F, we see that y = cor-
responds to u through y = B/A. In that case, we shall
say that y = oo corresponds to u through (16). With this
convention, every u for which A and B vanish has at least
one corresponding y.
There appear to be grounds for conjecturing that if u an-
nuls both A and B, and if more than one y corresponds to
w", then every analytic y corresponds to u.
CHAPTER IV
SYSTEMS OF ALGEBRAIC EQUATIONS
40. The preceding chapters contain, of course, a theory
of systems of algebraic equations in n unknowns, with ana-
lytic coefficients. One has only to suppose that the given
system 2 consists of forms which are of order zero in each
yi. But there are good reasons why algebraic systems should
receive special treatment.
To begin with, in most of the foregoing theory, algebraic
equations are forced into an artificial association with diffe-
rential equations. For instance, the closed essential irre-
ducible systems held by a system of forms of order zero in
each ?/i, are systems of differential forms. One does not
obtain thus Kronecker's theory of algebraic manifolds. We
shall see that a purely algebraic theory of algebraic systems
can be secured with the help of the notion of relative irre-
ducibility which was studied in 16.
But what is more important for us, from the standpoint
of differential equation theory, is that the theory of alge-
braic systems can be developed from the algorithmic point
of view, so that every entity whose existence is established
is constructed with a finite number of operations. The re-
sults of the algebraic theory, when applied to systems of
differential forms, will give us methods for determining the
basic sets of the irreducible systems in a decomposition of
a given finite system of differential forms. A theoretical
process will be given for obtaining equations which com-
pletely define the irreducible systems. Also, we shall be
able actually to construct the resolvents of irreducible systems
of differential forms.
62
IV. ALGEBRAIC SYSTEMS 63
Finally, we shall apply the theory of algebraic systems to
the study of the organic properties of the manifolds of systems
of differential forms.
Our results relative to algebraic systems are mainly con-
tained in the literature on algebraic manifolds.* For us, it
will be convenient, in deriving these results, to use the
methods of Chapters I and II.
INDECOMPOSABLE SYSTEMS OF SIMPLE FORMS
41. We define a domain of rationality to be, as in algebra,
a set of elements upon which the rational operations are
performable, the set being closed with respect to such
operations.t Every field is a domain of rationality.
Let a domain of rationality 3) be given whose elements are
functions of x, meromorphic in a given open region 21.
By a simple form, we mean a form in y^ , , y n which
is of order zero in each iji.% Wherever the contrary is not
stated, the coefficients in a simple form will be understood
to belong to 3).
A system 2 of simple forms will be said to be simply closed
if every simple form which holds 2 belongs to 2.
A system 2 of simple forms will be called decomposable if
there exist two simple forms G and H such that neither O
nor H holds 2, while QH holds 2. A system which is not
decomposable will be called indecomposable.
Every system 2 of simple forms is equivalent to a finite set
2 l , , 2 S of indecomposable systems. This is proved as in
13, and, in fact, is an immediate application of the results
* Macaulay, Modular Systems. Van der Waerden, Moderne Algebra,
vol. 2.
t The term ''domain of rationality" is being displaced, in common usage,
by the term "field". We have reserved the latter term for use as in the
preceding chapters. See Dickson, Algebras and their arithmetics, Chapter XI.
J We prefer this term to polynomial, since we shall have to use the
latter term in more general situations.
One can use here Hilbert's theorem on the existence of a finite basis
for every system of polynomials in n variables, in place of the lemma
of 7.
64 ALGEBRAIC DIFFERENTIAL EQUATIONS
on relative irreducibility in 16.* The decomposition is
unique in the sense of 14.
SIMPLE RESOLVENTS
42. Let 2 be any non-trivial simply closed system. Then
the unknowns can be divided into two sets, u\ , , u q and
2/i > -iVpj P + Q ni suc h that no non-zero form of 2 is
free of the y^ while, for j = 1, -, p, there is a non-zero
form of 2 in yj and the ui alone. We shall call the m a set
of unconditioned unknowns. Let the unknowns be listed in
the order
Ui , , itq ; y\ , , yp ,
and let
(1) A 19 ...,A P
be a basic set of 2. Each ^ introduces yt.
Then evert/ solution of (1) /or w/uc& fe initial of no Ai
vanishes is a solution of 2 . Furthermore, if 2 is indecomposable,
then (1) has regular solutions and every simple form which
vanishes for the regular solutions of (I) is in 23 These
facts are evident.
43. Let 2 be a non-trivial simply closed system. We are
going to show the existence of a simple form (?, free of the
yi and of a form
Q = lfi*/iH
where the Mi are simple forms free of the y^ such that, for
two distinct solutions of 2 with the same m (if m exist),
and with G^O, Q gives two distinct functions of x.
By a prime system, we shall understand a simply closed
indecomposable system.
Following 25, we consider the system of forms obtained
from 2 by replacing each yi by a new unknown Zi. We take
* To satisfy all formalities, one can take, as c?, the field obtained by
first adjoining to 3) the derivatives of all orders of the function sin 3),
and then forming all rational -combinations of the functions in the en-
larged set.
t The definition of regular solution is, of course, that of 23.
IV. ALGEBRAIC SYSTEMS 65
the system & composed of the forms of 2, the forms in the
Zi just described, and also the form
in which the A; are unknowns. Let A be any prime system
which & holds, and which does not contain every form yi *<.
We shall prove that A contains a non-zero form which in-
volves no unknowns other than the m and fa.
If A contains a non-zero form in the ut alone, we have
our result. Suppose that A contains no such form.
Since A has all forms in 2, A has, for j = 1, , p,
a non-zero form Bj in yj and the m alone. Then Ij the
initial of Bj, since it involves only the m, is not in A.
Similarly, let Cj, j = 1, , p be a non-zero form of A in
Zj and the m alone. Letting Zj follow the Ui in C/, we see
that the initial // of C; is not in A.
To fix our ideas, let us assume that y^ z is not in A.
Consider any solution of A for which
(2) (yi-*i)/i-"lpli'---/,!
does not vanish. For such a solution, we have
2 _ ^g (y ^2) H ----- h lp(y P * P )
AI
Viti
Let m be the maximum of the degrees of the Bj in the yj
and of the degrees of the Cj in the gj. Let be any positive
integer. We write, for s = 0, -,,
where F 5 is a simple form. Now, it is plain that, using the
relations Bj = 0, Cj = 0, we can depress the degree of
F* in each yj and in each gj to be less than m. The new
expression for each Af will be of the form
3* _ Ea
1 ~
66 ALGEBRAIC DIFFERENTIAL EQUATIONS
where D 8 is a product of powers of the 7/ and Ij. Let D
be the least common multiple of the D s . We write
s = 0, , , each H 9 being a simple form of degree less
than m in each y. and z.. Now the number of power products
of the y., #. of degree less than m in each y. and #. is w 2p .
Consequently, if we take a 2> w 2 ^, we can find a non-zero
polynomial in ^, of degree not greater than a, whose co-
efficients are simple forms in A 8 , *, Aj, and the M*, which
vanishes for every solution of A for which (2) does not
vanish. The form in the A*, m thus obtained belongs to A.
The existence of <? and Q is then proved as in 25. We
notice that, since we are dealing with simple forms, it is
possible to take the Mi here, which correspond to the ^ of
25, and to the Mi of 26, as integers ; in short, no derivatives
of the %i will appear in the forms K, L of 25.
44. Let 2 be any non-trivial prime system.
We take a pair (?, Q as in 43.
We introduce a new unknown w, and form a system A by
adjoining w Q to 2. Let ^ be the system of all simple
forms in w, the u i and y. which vanish for all solutions of A.
It is easy to prove, as in 28, that is indecomposable.
Those forms of 12 which are free of w are precisely the
forms of 2.
As above, we prove that has a non-zero form free of
the y t .
We now arrange the unknowns in Q in the order
u l9 "- 9 u q ; w\ y v --,y p
and take a basic set for &
(4) A , AI y , Ap .
Here, w, y l9 , y p are introduced in succession.
We take A algebraically irreducible relative to 3).
IV. ALGEBRAIC SYSTEMS 67
As in 29, it follows that each A t is linear in y i so that
the equation A i = expresses y i rationally in w and the Uj.
We call the equation A = a simple resolvent of 2 (or
of any system of simple forms equivalent to -2).*
It is easy now to prove that q, in 42, is independent
of the manner in which the m are selected.
BASIC SETS OF PRIME SYSTEMS
45. We consider simple forms in the unknowns
Let
(5) AI, A*, - - -, A p
be an ascending set of simple forms, each Ai being of class
q-\-i. We are going to find a condition for (5) to be a basic
set for a prime system.
In what follows immediately, we consider the m to be
complex variables, and the y i to be functions of the u i and x.
We represent Ai, with this interpretation of the symbols in
it, by at.
We denote by 93 an open region in the space of x\
MI, , u q , for every point of which x lies in 21.
We are going to prove that, for (5) to be a basic set of
a prime system, it is necessary and sufficient that
(a) Given any open region 93, there exist p functions, ', ", , (p)
of x; u ly - , u q , analytic in some open region contained
in 93, which make each 04 vanish when they are substituted
for yi, - , y p respectively, and
(b)/0r every i <p, given any analytic functions ',, C (i ~ 1}
of x\ Ui, , u q which cause i, , a t -i to vanish when
substituted for y l9 , yi i, the coefficient of the highest
power of yi in ai does not vanish for yj = t^, j ' 1,
, i 1 , and, after these substitutions, Oi, as a polynomial
in yi, is irreducible in the domain of rationality obtained
* One is equipped now to read the first part of Chapter VII.
68 ALGEBRAIC DIFFERENTIAL EQUATIONS
by adjoining to 3) the variables MI , , u q and the func-
tions?, --, (t '- 1) .*
Furthermore, we shall see that if (a) and (b) are fulfilled,
no non-zero polynomial in the m, yt, with coefficients in 3),
of lower degree than each a/ in yj, j = 1, , p can vanish
for yj f^, j = 1, , ^) where the t^are functions as in (a).
46. In 46, 47, we treat the necessity of the conditions.
We assume the existence of a prime system 2 for which (5)
is a basic set.
Let mi be the degree of at in yi.
Let 3> be the domain of rationality obtained by adjoining
the uj to 3). If 0,1 were reducible in 3) , ^ would be the pro-
duct of two forms, U and V, each of degree less than mi in y^.
As one of (7, F would hold -2, (5) could not be a basic set.
Then a\ is irreducible, so that the equation ai = deter-
mines y l9 in some open region 33i, contained in 23, as any
one of mi distinct analytic functions i, , m t of the
variables #; MI, , w ? .
As the coefficient of the highest power of y% in a 2 is of
lower degree in y l than a 1? that coefficient cannot vanish
identically in #; MI, , u q if y v is replaced by any J.
Let ^ be replaced by some & in a 2 and let # 2 be the
polynomial in y* which is thus obtained from a 2 . Let 3)* be
the domain of rationality obtained on adjoining the indicated
Cl to 3V
Suppose that 2 as a polynomial in y% , is reducible in 3)t .
Let 2 = ?>! y 2 with (pi and y a polynomials in y 2 , of positive
degree, with coefficients in 3)*. Each coefficient in y>i and $p 2
is of the form d/f, where d and /^ are polynomials in
u i9 - 9 Uq\ Ci with coefficients in 3).
Let be the product of all denominators ft. We may write
(6) 6a, = ViVt
* Thus, the enlarged domain of rationality is a set of functions of a?;
HI , , <, . For * = 1 , the above is to mean that a\ , as a polynomial
in 1/1, is irreducible when Ui, , w 7 are adjoined to 3).
The above result establishes an equivalence between the basic sets of
prime systems and certain sets of polynomials used by van der Waerden in
IV. ALGEBRAIC SYSTEMS 69
where ^ t and i/> 2 are polynomials in y 2 of degree less than w 2 ,
whose coefficients are polynomials in HI, , u q \ C{. Making
use of the relation i = for CJ we depress the degrees
in & of the coefficients in I//!, V> 2 to less than m l . Each
coefficient will be of the form y/y, with y a polynomial in
ui,-,Uq. Multiplying through in (6) by U the product
of the denominators y, one obtains a relation
(7) Ma a = ?ii
with ?i and ? 2 polynomials in t/ 2 of degree less than w 2 .
The coefficients of % l9 ? 2 are polynomials in {; MI, -, u?,
of degree less than m t in {. We notice that neither ?i nor J 2
vanishes identically in x\ ui, < t u q \ y*. Let t,g\jg* be
the polynomials which result respectively from ^0, ?i, J 2
on replacing J by y\. Then
(8) ta 2
vanishes for y l = t|.. Let s 1 be the coefficient of y^ in a x .
We obtain a relation
(9) s^(ta 2 g l g 2 )ka l = b
with k and 6 polynomials in MI, -, u qj y\, yz, and b of
degree less than mi in y^.
Now b vanishes identically in ui, , u^; 2/2 for yi = CJ.
Hence, if b is written as a polynomial in z/ 2 each coefficient
must vanish for y L = fi. As ai is irreducible in > and
as the coefficients in b are of degree less than m x in y lt
b must be identically zero.
Let Ji be the initial of A 1 , and let T, GI , G 2 , K be the
forms which t, g l9 g 2 , k become when the m, yt are regarded
as unknowns. Then
(10) R(TAt O l G*) KAi = .
We observe that GI and G 2 are reduced with respect to
A 19 A 2 and are not zero. Now (10) shows that l G 1 G 2
his treatment of prime ideals of polynomials. (Mathematische Annalen,
vol. 96, 1927, p. 189.)
70 ALGEBRAIC DIFFERENTIAL EQUATIONS
holds 2. Thus one of G l9 G 2 must hold 2, which is im-
possible.
Hence 2 is irreducible in 2V
47. Thus the equation 2 = defines y t as one of w 2
functions /', ^ = 1, -, w 2 , of a?; MI, -, u qj each analytic
in some open region $ 2 , contained in $1. Evidently we
can use a single open region 93 2 which will serve no matter
which & is used in determining the m 2 functions f". That
is, we have m, m 2 pairs &, j' which are solutions for ^i, 1/2
of i = 0, a t = and these m^ w 2 pairs are the only solutions
of ^ = 0, a 2 = analytic in 2S 2 .*
Consider any pair &, fj'. Let s 3 be the coefficient of the
highest power of y s in a 3 . We shall show that s s does not
vanish when y\ = &, 2/2 = t". Suppose that s 3 does vanish.
Let <5 be the polynomial in i/ 2 which ^ 8 becomes when y = K-
As 2 is irreducible in 3) x -, we see, since s 3 is of degree less
than w 2 in y% , that the coefficients of the powers of */ 2 in tf
are zero. That is, the coefficients of the powers of y t in s s
vanish for y = fj. But those coefficients are of lower degree
than mi in yi. This proves our statement.
Let ^/ be obtained from by the adjunction of } and C; r .
Let 3 be the polynomial in // 3 which a s becomes for z/ x = K,
2/a ==: ty. We shall prove that 3 , as a polynomial in y 3 ,
is irreducible in #.
Suppose that it is not. Then 3 = ^ ^ 2 with ^ and y> 2 poly-
nomials in 2/ 3 of degree less than m s , with coefficients in #.
Each coefficient in y t and y 2 is of the form d/ft, where rf and ^
are polynomials in z^, ,%; f{, fj', with coefficients in 3).
Let be the product of the denominators ft. Then
(11) Oa B = ^ i// 2
where Vi ^nd t// 2 are polynomials in y$ of degree less than iw s ,
whose coefficients are polynomials in MI, , M g ; C{, Cj'.
Making use of the relation 2 = for ff , we depress
each coefficient in ip l , i^ 2 to be of degree less than m 2
* The sets of m functions C}' corresponding to two distinct C< may
have functions in common.
IV. ALGEBRAIC SYSTEMS 71
in ". The new coefficients will be of the form Yin with
*l a power of the coefficient of yf* in 2 and y a polynomial
in M A , -, u q ; C{, j' of lower degree than m 2 in ". Thus
if 0j is the highest of the powers 17, we have
with f i , 2 polynomials whose degrees in y$ , / are respect-
ively lower than m B , w 2 . In the same way we depress the
degrees in J of the coefficients in 1 and ? 2 to be less
than Wi. We find thus
with 2 a power of the coefficient of yi in a l9 and with
r lr r g polynomials in w l7 -, u^; , j'; y 3 whose degrees
in 2/3, Cj r , i are respectively less than w 3 , tn 2 , m^. Further-
more, neither of r l9 r 2 vanishes identically in
Let ^, ^, ^2 result respectively from 2 t 0, TI and T 2 on
replacing K by y t and Cj' by y 2 . Then
vanishes for y { = fj, y 2 = /' Let ^ and 5 2 be respectively
the coefficients of y^\ in a x and of yf* in a 2 . Then we have
a relation
(12) Q sfr (ta 3 fft g 2 )h cii ks a 2 = I
with Jcij kz, b polynomials in M!, -, u q \ y lf y 2 , y$j the
degrees of b in y } and t/ 2 being less than mi and w 2
respectively.
Now b vanishes identically in w t , , u q m , y& if yi and y 2
are replaced by & and f" respectively. Hence, if 6 is written
as a polynomial in y B , each coefficient must vanish for
y\ = &f 2/2 == : fj'. Considering the degrees of the coefficients
in i/! and t/ 2 , we see by the argument used in proving that $5
does not vanish, that 6 vanishes identically. Thus (12) gives
a relation
72 ALGEBRAIC DIFFERENTIAL EQUATIONS
(13) /f' /* (TA B - G, 0.) - JKi ^ K 2 A, =
with 7 t - the initial of -4*, and with G\, G% not zero and
reduced with respect to AI, A 2 , A B .
This proves that 3 is irreducible in $/.
Continuing, w r e prove the necessity of the conditions
stated in 45.*
48. We turn now to the sufficiency proof. Let the con-
ditions stated in 45 be satisfied. We shall prove that (5)
has solutions for which no initial vanishes and that, if G
and H are simple forms such that GH vanishes for all so-
lutions of (5) which make no initial zero, then either G
vanishes for all such solutions, or else H does.
Let ', , (p} be functions as in 45. Then no 7; va-
nishes when the yi are replaced by the (i) . Let
(14) rr ; ?i, - - , % q
be values of x\ MI, -, u q for which the u) are analytic,
the coefficients in the Ai being analytic at a- and no Ii va-
nishing for the above values. If we take M* = &, i = 1, , </,
the (i) become functions of x which constitute a solution of
(5) with /! ... lp + 0.
Let now G and H be such that GH vanishes for all so-
lutions with /! Ip ^ 0. Let 6?i and HI be, respectively,
remainders for G and H with respect to (5). Then GI HI
vanishes for all solutions with /i I p ^ 0. Then (?i JBi
must vanish identically in #; 2/ 1? -, u q when the y t - are
replaced by the (i) as above. This is because, if the quan-
tities (14) are varied slightly, but otherwise arbitrarily, the
(l>) will still give a solution of (5) with 1\ - I p ^ . Hence
either GI or HI vanishes for the above replacements. Sup-
pose that GI does. Then Gt , being reduced wfth respect to
(5), vanishes identically. Thus G vanishes for all solutions
for which no initial vanishes, and we have our result.
* One might replace the ahove proof hy an induction proof, in which
irreducibility is proved for only one < with t > 1 . We think that, on
the whole, the ahove treatment is less oppressive than one by induction.
IV. ALGEBRAIC SYSTEMS 73
Let 2 be the totality of simple forms which vanish for
the solutions of (5) for which no initial vanishes. Then 2
is simply closed, and indecomposable. Now, if 2 contained
a non-zero form Cr, reduced with respect to (5), O would
vanish for yi u) , i = 1, , p. This is impossible.
Then (5) is a basic set for 2.
Of course, 2 contains no non-zero form in the m alone,
Also, by the methods of elimination frequently used, it can
be shown that, for j 1 , , p, 2 has a non-zero form
in yj and the m alone.
49. Let (5) be a basic set of a prime system 2. We have
seen that every solution of (5) for which no initial vanishes,
is a solution of -5". We shall now prove that every solution
of (5) for which no separant vanishes is a solution of 2.
Consider any solution of (5)
(15) M U , u q \ y l9 -, y p
for which no Si vanishes. Let G be any form in 2. Let
X Q be a point at which the functions in (15) and the co-
efficients in the Ai and O are analytic, and for which no
Si vanishes for (15). Let
be the values of (15) at # . An easy application of the
implicit function theorem shows that we can get functions
', , (p) as in 45, analytic at
(16) #o; 1, ,*
and assuming there the values ^ , . . , i\ p respectively.
If we put Ui = ut in C^, i = 1, , ^, t^ becomes yj.
Since the f O) make no /,- zero, we can find values
(17) xt; Ji,---,li
as close as we please to (16), which, with the corresponding
values i?i, , q' p of the ? 0) , make no 7* zero. If we take
Mi = ?J, i = 1, -, g, the S 0) give ^ analytic functions yj
which, with the Mi, constitute a solution of (5) for which
no initial vanishes. Hence G vanishes for
74 ALGEBRAIC DIFFERENTIAL EQUATIONS
if a?o is sufficiently close to x. By continuity, G vanishes for
This means that G f vanishes for (15). Our result is proved.
CONSTRUCTION OF RESOLVENTS
50. Before we can develop a method for the effective con-
struction of a resolvent for a prime system for which a basic
set is given, we must have a solution of the following problem.
Let A be a simple form in HI, , u q \ w, of positive
degree in w, irreducible as a polynomial in iv in 3> ( 46).
Let AI be a simple form in MI, , u q \ w\ y, of positive
degree in y. Let x be any analytic function of x\ MI, , u q
which renders A zero when substituted for w. Let a be the
polynomial in y obtained by replacing w by 1 in A. We
assume that the initial of AI does not vanish for w = 1-
It is required to determine the irreducible factors of in S>i,
the domain of rationality obtained by adjoining d to S) -
Several methods are known for resolving into its irre-
ducible factors. The following treatment is taken from
van der Waerden's Moderne Algebra, vol. 1, p. 130, where
a more general algebraic situation is considered.
It nrist not be thought that we must actually possess &
to carry out the factorization. It will be seen that all
operations used are rational, and that we get expressions
for the factors of a with no knowledge relative to f x except
that it renders A zero.
Let A be of degree m in w and let 2 , , m be the
analytic functions of x\ Wi, ,%, other than l7 which
render A zero when substituted for w. We assume all & to
be analytic in some open region 93.
Let z be an indeterminate and let & be the polynomial
in y and z which results on replacing y in a by y z f i .
Let A, i = 2, - , m result from A on replacing d by &>
i = 2, ..., m . Let y = A A &*
IV. ALGEBRAIC SYSTEMS 75
Then ^ is a polynomial in y, z with coefficients in , the
coefficients being capable of determination by the theory of
symmetric functions. Let y be resolved into irreducible factors
in o- This is possible, provided that we are able to factor
a polynomial in one variable with coefficients in .* Let
(18) Y = di-.-dr
with each di a polynomial of positive degree in y, z, with
coefficients in and irreducible in 3V Finally let r iy
i = i y . . . 9 r be the highest common factor of & and <$*,
both considered as polynomials in y, z, the domain of ratio-
nality being i . This highest common factor is obtained by
the Euclid algorithm, bearing in mind that a polynomial ?
in 1 , MI , , u q is zero when and only when the polynomial c,
in Wj obtained by replacing 1 by w, in ?, is the product
by A of a polynomial in w with coefficients in .
We shall prove that the highest common factors just found
become, for = 0, the irreducible factors of a in t . Let
a
be a resolution of into irreducible factors. Then
A = Vi V>2 Vk
where each ipi results from y>i on replacing y by y z^.
It is easy to see that each ^i> as a polynomial in t/, z, is
irreducible in < 3) 1 .
Manifestly each ^ is a common factor of A and of some
dj in (18). If we can prove that, in this case, ^ is the
highest common factor of A and (5;, we will have our result.
Let #^, for^" 2, , m, be the polynomial obtained
from V on replacing 1 by j. Let
(19) ^ = ^^^ m) .
Then 171 is a polynomial in t/, ^ with coefficients in 3) ? and
r = ?i ?* ?fc.
Each di in (18) is a factor of some iy;,
* Perron, Algebra, vol. 1, p. 210.
76 ALGEBRAIC DIFFERENTIAL EQUATIONS
Suppose, that Vi is a factor of d 1 and that d l is a factor
of f]i. If we prove that ty { is the highest common factor
of fli and //! we shall have our result.
Suppose, for instance, that ^ is divisible by ^ i/> 2 . Then,
by (19)
(20) v;'--- </T
where ^ is a polynomial in ?/, 0, with coefficients in j.
The set of terms of highest degree in T/, z in the first member
of (20) is of the form
(21)
with & a rational combination of the m and / The terms
of highest degree in the second member give an expression
of the type
(22) *(#,*) (y-*W.
Now (21) and (22) cannot be equal, since no y #& with
i>l is divisible by y z^. This completes the proof.
51. We consider a non-trivial prime system 2 in the un-
knowns Mi, i/i, for which
(23) AI, Av, , J^
is a basic set, each Ai introducing yt. In 53, 54 we show
how, when the At are given, a simple resolvent can be con-
structed for 2.
52. Let A t , , A p be new unknowns. Let 2 A be used to
represent 2 when the unknowns are the m, A, i/i. It is easy
to show, as in 17, that ^ is indecomposable. Furthermore,
no non-zero form in the ?(i, k holds 2\.
We see as in 43 (or 25), that there exists a non-zero
form O in the m, ^ such that, for two distinct solutions
of 2 t with the same m, k, for which G does not vanish,
the form
Q = ^i2/iH ----- \-ipyp
gives two distinct functions of x.*
* At present, we have no way of determining G.
IV. ALGEBRAIC SYSTEMS 77
By 44, a simple resolvent exists for 2 t , for which w = Q.
Let & be the system of all simple forms in the U{, h, w, yi
which hold 2 1 and w Q. We consider a basic set for &,
(24) R, B l9 -",B P
in which iv y yi, , y p are introduced in succession and in
which B is algebraically irreducible. Then B = is a re-
solvent for 2\ and each Bi is linear in yi.
53- We shall now show how a basic set (24) can actually
be constructed.
By the method of elimination of 31, we can determine,
by a finite number of rational operations, a non-zero simple
form 8 in w, the ^ and iii which vanishes for all solutions
of (23) and w Q for which no initial in (23) vanishes.
Then 8 belongs to &. Now, let
o ^^ &i op
with each Si algebraically irreducible relative to . Then
some Si holds . The selection of such an Si can be made
as follows. Consider any Si and let T be the form obtained
from it on replacing iv by Q. For Si to hold ., it is
necessary and sufficient that T hold 2 t . Let T be arranged
as a polynomial in the A t -. For Tto hold 2 l9 it is necessary
and sufficient that each coefficient in the polynomial hold 2.
A coefficient will hold 2 if and only if its remainder with
respect to (23) is zero.
Every form in the in, ^ and w which holds & is divisible
by B. Thus an irreducible factor of 8 which holds SI must
be the product of B in (24) by a function in 3).
We have then a method for constructing a simple resolvent
for 2 lt It remains to show how a complete basic set (24)
can be determined.
Let U be the form which results from B on replacing w
by w + yi and A x by Ai + 1. Then U holds . The degree
of U in yi is that of- B in w and the coefficient of the
highest power of y in U is free of ic.
78 ALGEBRAIC DIFFERENTIAL EQUATIONS
Now let represent any analytic function of the HI, Ii
and x which makes R vanish when substituted for w. Let
a be the polynomial in y^ obtained on replacing w in ?7by f .
Let
(25) cc = (*! a s <* m
be a decomposition of a into irreducible factors obtained as
in 50. The coefficients in the a* are rational combinations
of , the m and A$. Let /tf be the product of the denominators
of these coefficients. Then
The r are irreducible and their coefficients are polynomials
in , the M* and ^. Let B be the form which results from ft
on replacing by w. Let Ci result similarly from y<. Then
(26) BUCi C
m
vanishes identically in y t when w is replaced by . It follows,
as in 46, that (26) holds , hence that some d holds .
Suppose that Ci is found (by test) to hold &. We say
that Ci is linear in y lf If /! is the initial of RI in (24)
we have
(27)
where K is free of y lB As isT holds /2, it is divisible by
R. Thus, if Ci were not linear, (27) would imply that Y\
is reducible.*
It is only necessary, then, to take the remainder of Ci
with respect to R in order to have a form which will serve
as R l in (24).
The Ri with i>l in (24) are determined in the same
way,
54. It remains now to construct a resolvent for 2.
Let I be the initial of jR, in (24) and Ii that of Ri. As
Ii and R are relatively prime polynomials, we can find forms
Jlf,-, Nij Li such that
* We note that Ii cannot vanish when w is replaced by C.
IV. ALGEBRAIC SYSTEMS 79
(28) Mili + NiR = Li
with Li free of w, and not zero.
Let the A< be replaced by integers en in such way that
JL J-i J.p JL/I Lp
does not vanish. We shall show how (24) gives, for these
substitutions, a resolvent for 2 with
w = aiyi H ----- \~0pyp*
Let <D be the indecomposable system obtained by adjoining
w a l y l ----- a p y p
to 2. For the substitution A t - = a t -, (24) becomes a system
(29) #, R{, ..., 5;.
Then each form of (29) holds (Z>. Let R r be resolved into
its irreducible factors in . One factor $, which can be
determined, will hold <Z>.
If we put h = a* in (28), we see that no R'i has an
initial which is divisible by S. Let Si be the remainder of
R'i with respect to 8. Then each form in the set
(30) 8, &, , S p
holds 0.
It is clear that (30) is a basic set for the totality of
simple forms which hold CD. To show that S = is a
simple resolvent for 2, we have to prove the existence of
the form G of 43. If two distinct solutions of CD have
the same m and w, the m and w must make the initial
Ii of some Si in (30) vanish, Aus I'i and 8 are relatively
prime polynomials, we have a relation
MiH + NiS = Li
with Li a non-zero form in the m alone. Then ii L p ,
which can actually be constructed, will serve as G.
80 ALGEBRAIC DIFFERENTIAL EQUATIONS
RESOLUTION OF A FINITE SYSTEM INTO INDECOMPOSABLE SYSTEMS
55. Let 2 be any finite system of simple forms in y v --,y n ,
not all zero. In this section, we show how to determine
basic sets of a finite number of prime systems which form
a set of systems equivalent to 2. Later, we shall obtain
finite systems of forms equivalent to the prime systems.
Let
(31) ^,^,..-,4,
be a basic set of 2, determined as in 4. If A is of class zero,
2 has no solutions and is thus indecomposable. We assume
now that AI is not of class zero. For every form in 2,
let the remainder with respect to (31) be determined. If
these remainders are adjoined to 2, we get a system 2''
equivalent to 2. By 4, if not all remainders are zero,
2' will have a basic set of lower rank than (31). We see,
by 3, that after a finite number of repetitions of the above
operation, we arrive at a finite system A, equivalent to 2,
with a basic set (31) for which either AI is of class zero
or for which otherwise the remainder of every form in A
is zero.
Let us suppose that we are in the latter case. We shall
make a temporary relettering of the y^ If, in the basic
set (31) for A, A i is of class j t , we replace the symbol ^
by y r The q = n p unknowns not among the y. we call,
in any order, u i9 - , u q . We list all the unknowns in the
order u v , u q ; y v , y p .
With this change of notation, we proceed to determine,
using 45, whether (31) is a basic set for a prime system.
If A l is reducible, as a polynomial in y 1 and if A l = B l jB 2 ,
where J9 X and B 2 are of positive degree in y v then A will
be equivalent to A-\-Bi, A + B* and each of the latter
systems, after we revert to the old notation for the unknowns,
will have a basic set lower than (31).
Suppose then that AI is irreducible and let ' be any
analytic function of x\ Ui,-'-,u q which annuls AI when
substituted for y r Let <* 2 be the polynomial in y 2 which
IV. ALGEBRAIC SYSTEMS 81
AS becomes, for this substitution. Suppose that 2 is reducible
as a polynomial in ?/ 2 . By (10), there exist non-zero forms
Gi , G 2 , reduced with respect to AI , A s , such that /i (?i 6^
holds ^. Of course, 50 furnishes a method for actually
determining GI and O 2 . Then ^ is equivalent to
Each of the latter systems, after we revert to the old notation
for the unknowns, has a basic set of lower rank than (31).
This becomes clear if one considers that, in (10), T, Ot, G*, K
do not involve any ui not effectively present in AI and J 2 .
Suppose now that 2 is irreducible. Let A and A 2 vanish
for y^ = ', 2/ 2 = " Let 8 result in the usual manner
from A 8 . If # 3 is reducible with respect to t/ 3 , we see
from (13) that A is equivalent to
each of which latter systems, in the old notation, has a lower
basic set than (31). What we need, however, is a method
for resolving 3 into its irreducible factors. The irreducibility
properties of A l and 2 show that AI , A 2 is a basic set of
a prime system A 1 in u iy - , u ; y v y 2 . Let E -= be
a simple resolvent for A' , constructed as in 54, with
(32) w a l y l a^y 2 = 0,
#i, a 2 being integers. It is clear that ai' + a 2 " annuls E
when substituted for w, and that C' and f" are each rational
in aiC' + ait", with coefficients in . In short, if is
any form in the w<, yi and ?^, which holds A' and the first
member of (32), and if C results from B on replacing w by
then (7 holds A'. Then
so that C vanishes for y == f ; , y 2 = ". Thus, in factoring 8 ,
we may use the domain of rationality obtained by adjoining
82 ALGEBRAIC DIFFERENTIAL EQUATIONS
the \a and fliC' + OgC" to 3). The factorization is accom-
plished as in 50.
All in all, we have a method for testing A to determine
whether (31) is a basic set of a prime system, and for
replacing A by a set of systems each with basic sets lower
than (31) when the test is negative.*
Using now the old notation for the unknowns, let us
suppose that (31) ha,s been found to be a basic set for a prime
system. Let 2 t denote the latter system. Then A is equi-
valent to
Each A + Ii has a basic set which is lower than (31).
What precedes shows that the given system 2 can be
resolved into prime systems, as far as the determination of
basic sets of the prime systems goes, by a finite number of
rational operations and factorizations, provided that the
same can be done for all finite systems whose ibasic sets
are lower than those of 2. The final remark of 3 gives
a quick abstract proof that the resolution is possible for 2.
What is more, the processes used above, of reduction, factor-
ization and isolation of prime systems 2 l , give an algorithm
for the resolution.
56. It remains to solve the following problem: Given
a basic set
(33) 4i,---,4,
of a non-trivial prime system in yi, -, y n , each Ai being
of class q + i, (p + q = ri), it is required to find a finite
system of forms equivalent to .t
57- Let
(34) Zi = in yi + ---- Mm 2/n, i = 1 , , n,
*If, when the unknowns are Ui, , u q \ y l , , y p , (31) is a basic
set for the prime system 1*, then, when we revert to the old notation,
(31) will he a basic set for the system into which 12 goes.
1 2 of 55 leads to several systems & . For each 12 , we reletter the
unknowns appropriately. After finite systems are found, equivalent to
the various 12, we can revert to the original lettering.
IV. ALGEBRAIC SYSTEMS 83
where the Zi, tij are new unknowns. Given any g + 1 of
the Zi,
we find, by the method of 31, a non-zero form in those z\ and
the t\j which vanishes for arbitrary ty, provided that the Zi are
obtained, according to (34), from yi which satisfy (33) and
make no initial in (33) zero.
Let B be such a form in 2i, , 2^+1. Let m be the
degree of B considered as a polynomial in the Zi. We shall
show how to obtain a relation C == among Zi, , z q +\,
where C is of degree m as a polynomial in the z% and, in
addition, is of degree m in each Zi separately, i=l,
Let
(35) Zi = a,i*iH ----- ha*,<H-
where the z\ and the ay are new unknowns.
Then B goes over into a relation B' = 0, B' being
a polynomial in the z\ whose coefficients are simple forms
in the ty, ay. The degree of B' in each z\ will be effectively m.*
Furthermore, we can specialize the ay as integers, in such
a way that the determinant ay\ is not zero and that the
coefficient of the mth power of each z\ in B r becomes a non-
zero simple form in the ty. Let this be done, and let B"
be the form in the zt, ty into which B' thus goes.
From (34), (35), we find
(36) z( = rnyi-\ ----- h*tny, * = 1, , 2+1,
where each T^ is a linear combination, with rational numerical
coefficients, of the ty with i<, q + 1 . From (35), (36), we
see that the ty with i^q+1 are linear combinations of
the Ty, with integral coefficients. Hence, the Ty may be
made to become arbitrarily assigned analytic functions, if
the ty are taken appropriately.
In the relation B" = 0, we substitute, for each ty, its
expression in terms of the Ty. Then B" goes over into a
* Perron, Algebra, vol. 1 , p. 288.
84 ALGEBRAIC DIFFERENTIAL EQUATIONS
form B'" in the g' l9 Ty, i = 1, , g+1. We now replace,
in B u> ', each T^ by fy- and each zj by z%. Then '" goes
over into a form C in *i, , ^1-1 and the ^/, (7 being of
degree w as a polynomial in the z\, and of degree m in
each *j separately. Furthermore C vanishes for all zi given
by (34) for which the yi satisfy (33) and make no initial
zero. This is because (36) may be considered equivalent to
the first q + l relations (34).
Evidently the relation (7=0 will subsist if we replace
z\, , z q \.\ by any <y+l of the z^ provided that a corre-
sponding substitution is made for the ty in (7.
We now specialize the t l j in (34) as integers with a non-
vanishing determinant, in such a way that the relations
obtained from C for the various sets of g+1 unknowns
remain of effective degree m in each zi appearing in them.
These relations will have coefficients in 5).
58. We consider z\, , z n with the ty fixed as above.
If the y t are replaced in (33) in terms of the z^ we get
a system & of p forms in the Zi. Let basic sets be deter-
mined for a set of prime systems equivalent to CD. Let
2 i9 , 2 S be those prime systems which are not held by
the initial of any At in (33), the yi being replaced in the
initials in terms of the **.* There will be one of the 2i which
holds the remaining 2 Z . This is because, in a resolution
of (33) into indecomposable systems none of which holds
any other, there is precisely one which is held by no initial.
To determine that - t which holds the others, all we need
do is to find a ^ whose basic set holds the other 2 iu
Suppose, for instance, that the basic set of ^ holds
2 2 , ..-, 2 a . Then, if 2 l does not hold 2j, the initial of
some form in the basic set of 2 l must hold 2j. Then
surely 2j cannot hold 2 lt Thus if 2 does not hold all J t -,
no 2j can hold all 2 t . Then ^ holds all 2 if *
2i is obtained from /2, ( 56) by replacing the yi in terms
of the Zi. We shall prove that 2 l9 like fi, has q unconditioned
* The condition for a form to hold a prime system is that its remainder
with respect to the basic set vanish.
IV. ALGEBRAIC SYSTEMS 85
unknowns ( 42). To begin with, it is easy to see that the
forms in any #+1 of the z t -, found in 57, belong to 1\.
On the other hand, if there were fewer than q unconditioned
unknowns in 2 1 , we could use the basic set of ^i to determine
a non-zero form in yi, , y q belonging to -Q.
Changing the notation if necessary, let zi, , z q be un-
conditioned unknowns for 2 lt Then 2 l will have a basic
set
(37) S l9 ...,B P
in which BI introduces z q +i. We assume, as we may, that
BI is algebraically irreducible.
59- We construct a simple resolvent R = for ^ , with
(38) w = rtiZ0.t-H ----- \-aptq-i-p9
the ai being integers. Let R be of degree g in iv.
We shall prove that the initial of R is a function of x
in . According to 57, each z, ?>g, satisfies with
1, ? 2V/ an equation of degree w in zj ? the coefficient of zf
being a function of x in 5). We may and shall assume that
the coefficient of z? 1 , i>#, in each of these p equations is
unity. Then (38) shows that w satisfies with zi, , z q an
equation in which the coefficient of the highest power of w
is unity.* This implies that, in the algebraically irreducible
simple form R, the coefficient of w g is free of zi, , z g .
We may and shall assume that the coefficient of w g is unity.
We shall show that
/QQ x Eio + Emv+.
(39) Zi = ------- - -- -
i = g-f 1 ? . . ., 71, where the Ey and D are forms in zi, -, z q .
Let
Mz q +i N = 0,
where Jtf and N are forms in w\ zi, , z g of degree less
than g in w. As jlf and J2 are relatively prime polynomials,
we have
* This is analogous to the fact that the sum of several algebraic integers
is an algebraic integer. See Landau, Zahlentheorie, vol. 3, p. 71.
86 ALGEBRAIC DIFFERENTIAL EQUATIONS
PM+QR = L
where L is a non-zero form in zi y , z q . Then
Lz q+ iPN = 0,
and, replacing PN by its remainder with respect to R, we
have a relation (39) for z q +i. Evidently we may use the
same I) for ^+1, , z n <
60. Let MI, , Up\ v be new unknowns and let -A be
the totality of simple forms in the gi, ut and v which hold
2 i and
V - UiZq+i ----- U p g n *
Then A has an algebraically irreducible form Zin v, zi, '-,z q
and the m, the coefficient of whose highest power of v is
unity.*
We shall prove that Z is of degree g in v. Using (39),
we see that
where the K* are simple forms in Zi, , z q and the w*, and
where w is given by (38). For the first g powers of v, we
get similar expressions if we make use of R = 0. We in-
fer, by a linear dependence argument, that v satisfies, with
z\j , z q and the Ui, if Z)^0, an equation of degree at
most g in v. The condition that i> 4" is removed by con-
sidering that A is prime. Thus Z is at most of degree g in v.
On the other hand, as v becomes w if w* = a/, i = 1 ,,#,
Z cannot be of degree less than # in v.t
Let v be replaced in Z by
(40) Mi*f+iH ----- I-WP**.
Then ^ becomes a form in *!,, * n and the w. Let this
form be arranged as a polynomial in the m with coefficients
which are forms in the 5 2* ,
* Note that each Ui z q +t satisfies an equation with the coefficient of
the highest power of M<Z,+I equal to unity.
f Note that, since the coefficient of the highest power of t; is free of
the M<, Z cannot vanish identically for u< = a*.
IV. ALGEBRAIC SYSTEMS 87
Let ^ be the finite system of these coefficients (forms in
the Zi). We are going to prove, in the following sections,
that ^ is equivalent to 2 lt Thus, if the Zi are replaced in
W by their expressions (34), we get a finite system of forms
equivalent to .. We shall thus have solved the problem
stated in 56.
61. We begin with the observation that for given analytic
functions z , , z n to constitute a solution of *F, it is
necessary and sufficient that, for v as in (40), and for
*i, , ss q as just given, Z vanish for arbitrary m. This
shows, in particular, that ^ holds 2.
Let Q be the discriminant of R with respect to w and let
K = DO,
where D is as in (39). We shall prove that every solution
of W with K ^ is a solution of 2^ . Let 5 X , - , n be
such a solution of W. Corresponding to ? 1? , % q , the
equation B = gives g distinct solutions for w. Using
each such w in (39), we get g distinct solutions
& ... *o') . . . *O) ?* i . . . /T
si, , <? g , ^4-1 > > ^ n > J - 1 > > y
of ^i. Let >J be the polynomial which Z becomes for Zi ,-,
i = 1 , - , q . Then
But v MiJg+i u p %n is a factor of ft. This shows
that for some j, gfJ> = &, i = q-}- 1, - -, n, and proves
our statement.
62. We are going to show that, given any solution 1, ,
of W, the ?i being analytic in some open region 2li in 21,
there exists an open region 21', contained in 2l t , in which
the solution can be approximated uniformly by a solution of W
for which, throughout 21', K^O. That is, for every *>0,
there exists a solution ^ , , ^ of V, analytic throughout 21',
such that IT + throughout 2f and that | & ^< | <* through-
out 21', f = 1, , n.
7
88 ALGEBRAIC DIFFERENTIAL EQUATIONS
This will show that 2i holds f, for since a form in 2
vanishes for every solution of *P with K^ 0, it will vanish,
by continuity, for every solution of W. We shall thus know
that *P and 2 are equivalent.
63. We shall establish the more general result that if H
is any non-zero simple form in * t , ,<?, then given any
solution of W analytic in 3d , there is an 2f in 21! in which
the solution can be approximated, as above, by a solution
of W with H distinct from zero throughout 21'.
It will evidently suffice to consider a solution of W for
which H=0.
We assume 2l x to be so taken that the equations of degree m
which 3+1, , z n each satisfy with gi, - - , z q ( 59) have
their coefficients analytic throughout 21 A . We assume also
that the coefficients in H, in K of 61, in R of 59 and
in D and the Ey of (39) are analytic throughout 2l x .
There is evidently no loss of generality in assuming that
H is divisible by K. We make this assumption.
Let 61 , , b q be constants such that
does not vanish for every x. Then, if h is a complex variable,
(41) #(?1+M, ,, + M)
is a polynomial in h of the type
(42) a r h r -\ ----- \-<* s h 8 ,
where the < are functions of x analytic in 2li. Since (42)
vanishes for h = 0, we have r 2> 1. We assume that a r is
not zero for every x.
Let 21 2 be an open region in 2li in which a r is bounded
away from zero. Let h be small, but distinct from zero.
Then (42) cannot be zero at any point of 21 2 . Thus, if
(43) * = & + M, * = 1, ,,
jg = o will have g distinct solutions for w, each analytic
in S1 2 . This is because H is divisible by the discriminant of E.
IV. ALGEBRAIC SYSTEMS 89
As H is divisible by D in (39), [Si will have g distinct
solutions with Zi, , z q as in (43), for which z q +\ 9 , z n
are given by (39) and are analytic throughout 21 2 .
Consider a sequence of non-zero values of h which tend
towards zero,
(44) hi,hi,-..,Jn,-..
each hi being so small that (42) is distinct from zero through-
out 21 2 . For eacli 7^, if
(45) zj = Zj+l)jhi, ^ = 1, ,?,
Z will vanish if
(46 ) i; = Ui ^ i+ ... +Up ^
A; =1, ,#, where the zf } are analytic throughout 21 2 .
It is understood, of course, that the sf depend on 7i r For
any hi, the g expressions (46) are distinct from one another.
As the equation of degree m which each Zj, j>q satisfies
with z x , - -, z q has unity for the coefficient of zj 1 , ( 59),
there is a region 21 8 in 2I 2 and a positive number d such
that, throughout 21 3 ,
(47) zf | < d
for j = q + 1, -, n\ k = 1, , # and for every In in (44).
This is because the coefficients of zj 1 " 1 , -, gfj, in the above
considered equation, are bounded quantities.
For each In of (44), let one of the g expressions (46) be
selected, and designated by v (i> . We form thus a sequence
(48) t/, *", ---, !/, -...
Let 21' be any bounded open region which lies, with its
boundary, in 21 8 . From (47) we see, using a well known
theorem on bounded sequences of analytic functions,* that,
for some subsequence of (48), the coefficients of each m,
i=l y ... j p ? converge uniformly throughout 21' to an analytic
function. Let the limit, for the subsequence, of the coefficient
of m be {. We find thus that if
* Montel, Les families normales de -fonctions analytiques, p. 21.
Dienes, The Taylor Series, p. 160.
7*
90 ALGEBRAIC DIFFERENTIAL EQUATIONS
(49) *, = &, j = l,
Z vanishes for
Deleting elements of (44) if necessary, we assume that the
convergence occurs when the complete sequence (48) is used,
rather than one of its subsequences. For each hi there are
g 1 expressions (46) not used in (48). Let one of these
g 1 expressions be selected for each In, and let (48) be
used now to represent the sequence thus obtained. As above,
we select a subsequence of (48) for which the coefficients
of each m converge uniformly in 21'. This gives a second
expression
V = Ml
which causes Z to vanish when (49) holds. Continuing, we
find g expressions
(50) t; = l CS. 1 +...+t f pC5f ) >
k = 1, . ., g, which make Z vanish when (49) holds.
Let v k represent the second member of (50). Again, let
Wk represent the second member of (46), it being understood
that the subscripts k are now assigned, for each hi, in such
a way that the coefficient of in in Wk converges to that
in Vk as hi approaches 0.
Then since the g expressions Wk are distinct from one an-
other for every h i9 we will have, representing by ft the
polynomial which Z becomes when (45) holds,
ft = (v wi) - - (v w g ) .
By continuity, if we represent Z, when (49) holds, by y,
Y = (v vi)-- (vvg).
But since 1 , , J n is a solution of *F,
V Ui $ q +i ----- Up % n
must be a factor of y. This shows that, for some &,
IV. ALGEBRAIC SYSTEMS 91
This establishes the result stated at the head of the present
section and proves that *P is equivalent to 2 lt
A SPECIAL THEOREM
64. We prove the following theorem.
THEOREM: Let 2 be an indecomposable system of simple forms
in y , , y n Let B be any simple form which does not hold 2.
Given any solution of 2, analytic in an open region 2li, there is
an open region 21', contained in 2li, in which the given solution
can be approximated uniformly, with arbitrary closeness, by
solutions of 2 for which B is distinct from throughout 21'.
We assume, without loss of generality that 2 is prime.
If the transformation of 57 is effected, 2 may be replaced
by JSi, while B goes over into a form C in ^, , z n .
C does not hold -Si. Let g-H, ># be replaced in C
by their expressions (39). We find that, for all solutions
of ^i , with D ^ ,
fxi \ ri ^
W A ; ^ ~f)t* '
where N is a simple form in w] Zi, - , z q . In (51), w is
supposed to be given by the second member of (38). Because DC
does not hold J5i, Ni& not divisible by R of 59. Thus, we have
(52) XE+YN = H,
where H is a non-zero simple form in Zi, - - -, z q .
Let 21 2 be a region, contained in 2li, in which the coef-
ficients of the forms in (51) and (52) are analytic. We see
that if a solution of J5i makes C vanish at some point c
in 21 2 , then DH vanishes at c. But there is a region 21'
in 21 2 in which any given solution of J5i can be approximated
uniformly by a solution for which DH, hence C, is distinct
from zero throughout 21'. As the yi vary continuously with
the Zi, we have our theorem.*
*This useful theorem, and the considerations which lead up to it, do
not seem to exist in the literature, even for the case of equations with
constant coefficients. Professor van der Waerden recently communicated
to me a different proof, which deals with the case of constant coefficients.
CHAPTER V
CONSTRUCTIVE METHODS
CHARACTERIZATION OF BASIC SETS OF IRREDUCIBLE SYSTEMS
65. Let
(1) AI, A*, - - -, Ap
be an ascending set of differential forms in
UL, , Uqi y\_ , , yp ,
each Ai being of class q + i. We are going to find a necessary
and sufficient condition for (1) to be a basic set of a closed
irreducible system.
Let the order of At in yi be n. We represent y^ by Zi,
i = 1 9 . . . 7 p. The remaining ytj in (1) and the ny present
in (1), we designate now by symbols vie, attributing the sub-
scripts k in any arbitrary manner. With these replacements
of symbols, (1) goes over into an ascending set of simple
forms,
(2) B 19 B 9 , -, JB P
in the unknowns
(3) V , , V r \ 1, ' ' ' , Zp*
The passage from (1) to (2) is purely formal. Once it is
effected, we treat (2) like any other set of simple forms in
the vi, Zi. For instance, whereas, in a solution of (1), yij+\
must be the derivative of i/</, any set of analytic v<, Zi which
annul the Bt will be considered a solution of (2).
We are going to prove that for (I) to be a basic set of
a closed irreducible system, it is necessary and sufficient that
(2) be a basic set for a prime system in the unknowns (3),
the domain of rationality being 3 r .
92
V. CONSTRUCTIVE METHODS 93
We prove first the necessity. Suppose that the condition
is not fulfilled. Referring to 55, and also 46, 47, we
see that, since (2) is not a basic set of a prime system,
there exists, for some j, an identity
jft . . . jfr (TB. - &W -K.B, ----- K^Bj,, = 0,
where Ii is the initial of Bt and <?i , O t are non-zero forms in
the unknowns in JBi, , B/, which are reduced with respect
to #!,...,,.
To this identity, there corresponds an identity in forms in
the m, yi,
(4) Jp-J
where J* is the initial of Ai. Here H and H 2 are non-zero
forms of class q-}~j, which are reduced with respect- to
AI, , Aj. Thus, if (1) were a basic set of a closed ir-
reducible system, either H , jff 2 or some Ji would belong to
the system. This completes the necessity proof.
Suppose now that the condition is fulfilled. We shall prove
first that (1) has regular solutions. Consider any regular
solution of (2). Let a be a value of x for which the func-
tions in the solution, and the coefficients in (2), are analytic,
and for which no initial or separant in (2) is zero (for the
given solution). Let the values of the Vi, z* at a be assigned
to the corresponding w#, yg in (1). We construct functions
M!, , Uqj analytic at a, for which the w</ in (1) have the
indicated numerical values. If we assign to the first r\ 1
derivatives of yi, at a, the numerical values associated with
them above, the differential equation A t = will have a
regular solution in which the Ui are the functions above and
in which the first n derivatives of t/i have, at a, the above
assigned values. We substitute MI, , u q \ y into A% and
solve for y 2 with the initial conditions determined above.
We have now a regular solution of AI , At . Continuing, we
find a regular solution of (1).
Now, let G and H be two forms such that O H vanishes
for all regular solutions of (1). Let (? x be the remainder
94 ALGEBRAIC DIFFERENTIAL EQUATIONS
of O with respect to (1), and HI the remainder of H. Then
GI HI vanishes for all regular solutions of (1). It may be
that GI and HI involve certain mj not effectively present
in (1). In that case, let new symbols vt be added to (3)
for the new M#. Then (2) will be a basic set for a prime
system even after this adjunction of unknowns, for it will
continue to satisfy the condition of 45.
As we saw above, the values of the functions in a regular
solution of (2), at a point a which is quite arbitrary, are
values of the uij, yij in a regular solution of (1). This
means, if G 2 and H 2 are obtained from GI and HI by
replacing the Uij, yg by the vt, zi, that G 2 H 2 vanishes for
all regular solutions of (2). Hence either G 2 vanishes for
all regular solutions of (2) or H 2 does. Suppose that G 2 does.
As G 2 is reduced with respect to (2), G 2 vanishes identically.
Then G vanishes for every regular solution of (1).
Thus, the totality 2 of forms which vanish for the regular
solutions of (1) is an irreducible system. What precedes
shows that if a form G belongs to 2, the remainder of G
with respect to (1) is zero. This means that 2 has no non-
zero form reduced with respect to (1), so that (1) is a basic
set of 2. The sufficiency proof is completed.
66. We shall prove that if (1) is a basic set of a closed
irreducible system 2, then every solution of (I) for which no
separant vanishes is a solution of 2.
Let Si be the separant of At. Let G be a form which
vanishes for all regular solutions of (1). As in 5, we can
show the existence of integers $1 , , s p such that, when
a suitable linear combination of derivatives of AI, , A p ,
with forms for coefficients, is subtracted from
the remainder, O l9 is not of higher order than any Ai in yi,
= !,..., p.
Let H result from GI when we pass to the unknowns (3).
Then H vanishes for every regular solution of (2). Hence,
by 49, H vanishes for every solution of (2) for which no
V. CONSTRUCTIVE METHODS 95
separant vanishes. Then GI vanishes for every solution of
(1) for which no separant vanishes. So does (7. This proves
our statement.
BASIC SETS IN A RESOLUTION OF A FINITE SYSTEM
INTO IRREDUCIBLE SYSTEMS
67. Let 2 be any finite system of forms in yi 7 , yn,
not all zero. In this section, we show how to determine
basic sets of a finite number of closed irreducible systems
which form a set of systems equivalent to 2. In Chapter VII,
we give a theoretical process for determining finite systems
equivalent to the closed irreducible systems.
Let
(5) AI, A 2j - - -, A p
be a basic set of 2, determined as in 4. If A ly is of
class zero, ^ has no solutions, and is thus irreducible. We
assume now that AI is not of class zero. For every form
in 2, let the remainder with respect to (5) be determined.
If these remainders are adjoined to 2, we get a system 2'
equivalent to 2. By 4, if not all remainders are zero,
2' will have a basic set of lower rank than (4). We see,
by 3, that after a finite number of repetitions of the above
operation, we arrive at a finite system A, equivalent to 5,
with a basic set (5) for which either AI is of class zero or
for which, otherwise, the remainder of every form in A is zero.
Let us suppose that we are in the latter case. We shall
make a temporary relettering of the yi. If, in the set (5)
for Ay At is of class ji, we replace the symbol yj t by yi.
The q = n p unknowns not among the yj t , we call, in
any order, MI, , u q . We list all the unknowns in the
order ti, , u q \ yi, , y p .
With this change of notation, we determine, by 65,
whether (5) is a basic set of a closed irreducible system.
If it is not, we see from (4), that A is equivalent to
Each of the latter systems, after we revert to the old notation,
will have a basic set lower than (5).
96 ALGEBRAIC DIFFERENTIAL EQUATIONS
If when the unknowns are the m, yt, (5) is a basic set
of a closed irreducible system ., then, when we revert to
the old notation, (5) will be a basic set for the closed
irreducible system into which Q goes.
Using now the old notation for the unknowns, let us suppose
that (5) has been found to be a basic set for a closed
irreducible system. Let ^ denote the latter system. Then.
by 66, A is equivalent to
Each sl + Si has a basic set which is lower than (5).
What precedes shows that the given system 2 can be
resolved into irreducible systems, as far as the determination
of basic sets of the irreducible systems goes, by a finite number
of rational operations, differentiations and factorizations,
provided that the same can be done for all finite systems
whose basic sets are lower than those of 2. The final remark
of 3 gives an abstract proof that the resolution is possible
for 2. What is more, the processes used above give an
algorithm for the resolution.
In the resolution into irreducible systems obtained above,
some systems may be held by others.
The algorithm obtained above contains in itself a complete
elimination theory for systems of algebraic differential equations.
We get all of the solutions of 2 by finding the solutions of
each basic set which cause no separant to vanish. A solution
of an irreducible system which annuls some separant will be
a solution of some system like A -f fli above, and hence will
ultimately be found among the solutions of some other
irreducible system, where it annuls no separant. Thus our
algorithm reduces the process of determining all solutions of
a system of algebraic differential equations to an application
of the implicit function theorem and of the existence theorem
for differential equations.
It follows from what precedes that a system of forms in
yi> > l/ny in which each form is linear in the yij, is an
irreducible system.
V. CONSTRUCTIVE METHODS 97
TEST FOR A FORM TO HOLD A FINITE SYSTEM
68. Let CD be any finite system of forms. Let it be
required to determine whether a given form holds .
What one does is to resolve <D into irreducible systems as
in 67. For G to hold <Z>, it is necessary and sufficient
that hold each irreducible system. The condition for G
to hold one of the irreducible systems is that its remainder
with respect to the basic set of the irreducible system be zero.
This gives a test which involves a finite number of steps.
CONSTRUCTION OF RESOLVENTS
69. Let
(6) A\, A 2 , , A p ,
where the At are forms in w u , u q \ y , - -, y p , each Ai of
class q-\-i, be given as a basic set of a closed irreducible
system J. We suppose that either ^ does not consist entirely
of constants, or Ui actually exist.
We shall show how to construct a resolvent for 2.
We begin by showing how to obtain the form G of 25.
Let Bi be the form obtained from At, by replacing each yj by
a new unknown zj. We consider the finite system com-
posed of the forms of (6), the forms
(7) B 19 --.,B P
and also
^1(2/1 *iH l-ipfyptp),
where the A* are unknowns. We order the unknowns as follows:
Ui j ' ' ' > U q ; /! , - - , /, p ; y , , y p ; z\ , , Zp .
We apply the process of 67 for resolving Q into irreducible
systems, each irreducible system being represented by a basic
set. The theory of 25, 26 shows that each irreducible
system which is not held by every form yi Zi has a basic
set containing a form in the Ui and A alone. We obtain,
by a multiplication of such forms, the form K of 25.
When & contains a non-constant function, the determination
of (AI which do not annul L of 25 is an elementary problem
98 ALGEBRAIC DIFFERENTIAL EQUATIONS
whose solution is sufficiently indicated in 25. When m exist,
we find the Mi of 26 by inspection.
To avoid tedious discussions of notation, let us limit
ourselves now to the case in which $ does not consist of
constants. Consider the system
(8) Ai,---,Ap, w (^y v -\ ----- \-t* P y P )
in the unknowns
(9) M U -, u q \ yu , y p \ w.
The totality of forms which vanish for all solutions of (8)
which annul no separant is the system fi of 28. Every
other closed essential irreducible system held by (8) is held
by some separant.
We rearrange ihe unknowns (9) in the order
MI, -, u q \ w\ yi, -, y p ,
and apply the process of 67 to resolve (8) into irreducible
systems. We test these irreducible systems to see whether
they are held by the separant of some Ai, and pick out
those, say JSi, , JS" 5 , which are held by no separant.
As (8) has only one essential irreducible system which is
held by no separant, there must be one Si which holds all
other Si. To find such a Si, we need only find a Si whose
basic set holds all other Si. For, let the basic set of Si,
hold S 9 , - , 2 8 . If 2\ does not hold Sj, the separant of
some form in the basic set of Si, must hold Sj, so that Sj
cannot hold 2 l% Thus, if ^ does not hold every Si, no
Sj can hold every -,.
Si will have a basic set
in which R is an algebraically irreducible form in w and
the HI and in which JRi, /= 1, ,/>, introduces yi. By
28, 29 each equation J? = determines yi rationally in w
and the m and R = is a resolvent for S.
V. CONSTRUCTIVE METHODS 99
A REMARK ON THE FUNDAMENTAL THEOREM
70. The results of 65-67 furnish a new proof of the
fact that ever} 7 finite system of forms is equivalent to a finite
number of irreducible infinite systems. Using the lemma of
7, we obtain the theorem of 13. This new proof of the
fundamental theorem appears to us not to depend on Zermelo's
axiom. But only that part which is stated above has been
demonstrated on a genuinely constructive basis.
JACOBI-WEIERSTRASS CANONICAL FORM
71. Let 2 be a closed irreducible system with (1) for basic
set. Let A be the prime system for which (2) is a basic set.
We build a simple resolvent, R = 0, for ^, with
w #1 1 - a p z p 0,
the en being integers. We have
(10) MiZi N- t i=i,... 9 p 9
where the MI, Nt are simple forms in w and the v<.
Let & be the totality of forms which vanish for the common
solutions of 2 and
Let R go over into a form R' when the vt, Zi, are replaced
by the corresponding u^, y^. Similarly, let (10) go over into
(11) M' i y tr -N' i = 0.
Then R' and the first members of (11) are in . It can be
shown that 2 consists of all forms in the u i9 y t which vanish
for all solutions of (11) and R' = for which the separant
of R' and the Ml do not vanish.
Suppose that there are no m. In that case, the system (11),
with w defined by R' = 0, when converted into a system
of the first order, by the method of adjunction of unknowns
used in differential equation theory, assumes a form equivalent
to the Jacobi-Weierstrass canonical form.*
*Forsythe, Treatise on Differential Equations, vol. U, pp. 11-14.
CHAPTER VI
CONSTITUTION OF AN IRREDUCIBLE MANIFOLD
SEMINORMAL SOLUTIONS
72. Let 2 be a non-trivial closed irreducible system in
V' 2/i ' * ' ^ for which
(1)
each .4 t of class q-\-i, is a basic set.
A solution of (1) for which no separant vanishes will be
called a normal solution of (1). By 66, every normal
solution of (1) is a solution of -2.
A solution
(2) HJ,---,^; y 19 ..-,y p
of (1) for which some separant vanishes will be called
seminormal if there exists a set of points, dense in the
area 33 in which the functions in (2) are analytic, such that,
given any point a of the set, any positive integer m, and
any *>0, there exists a normal solution of (1), u 19 - - , y p >
analytic at a, such that
I Mffcfo) HI* (a) | < c , ! yjk(a) yjk(a
i = 1, , q\ j = 1, -, p\ k = 0, , m.
The results of this section and of 73 will show that the
existence of a single point a, as above, implies the existence
of a set of such points dense in 39. That is, a solution for
which some separant vanishes, and for which a single point a
exists, is a seminormal solution.
We shall prove that if O is a form until coefficients mero-
morphic in 21, the coefficients not belonging necessarily to ^,
100
VI. CONSTITUTION OF MANIFOLDS 1Q1
and if G vanishes for all normal solutions of (1), then G
vanishes for all seminormal solutions of (1).
More generally, we shall show that G vanishes for every
solution (2) for which a single point a, as above, exists.
Multiplying G by a power of (x a), if necessary, we assume
that the coefficients in G are analytic at a. When (2) is
substituted into (?, G becomes a function y(x) of x which is
zero at a. This is because G vanishes for all normal solutions
and because of the w, e property of a. Similarly, <p'(x)
vanishes at a, because the derivative of G vanishes |or every
normal solution. In the same way, every derivative of y (x)
vanishes at a, so that y>(x) is identically zero, and G vanishes
for (2).
If we restrict ourselves to forms G with coefficients in e?,
we see that every seminormal solution of (I) is a solution of 2 .
73* We are going to prove that the manifold of 2 is
composed of the normal solutions of (1) and of the seminormal
solutions.
In particular, the general solution of an algebraically ir-
reducible form A is composed of the normal solutions of A and
of the seminormal solutions.
Let At be of order n in yi. Let 6V be the separant of At.
For every yi s with s>n, in a normal solution of (1), we
have an expression
(4) yis = F
where B is a form of class at most q-{-i and of order at
most TJ in i/;, j = 1, -, i, and where F is a product of
powers of 5i, ,&. The forms
(5) Fy is B
belong to 2.
Let m be any integer greater than every r*. We adjoin
to (1) all forms (5), for i = 1 , , p, with * <; m. Without
going through the formality of replacing the u^ yij by new
symbols, let us consider the forms in (1) and (5) as a system Q>
of simple forms in the ?#, yy. That is any set of analytic
102 ALGEBRAIC DIFFERENTIAL EQUATIONS
functions Uij , ?/;;, which annul the forms of <Z> will be a solution
of <Z>. We do not ask, for instance, that yij+i be the derivative
of .?</
We shall prove that the totality & of simple forms which
vanish for all solutions of <Z> for which no Si vanishes, is
a prime system. Let GH vanish for all solutions of <P which
annul no Si. By (5), we have, for these solutions,
where B t and B 2 involve no yy with j > n and where JFi
and jP 2 are power products of the $. Then JB t J5 3 vanishes
for the above solutions.
By 65, (1), regarded as a set of simple forms, is the
basic set of a prime system (even after new mj are introduced).
Then either B l vanishes for all solutions of the simple forms
(1) which annul no Si or i? 3 does. Suppose that BI does.
Then O vanishes for all solutions of <D which annul no
separant so that tt is prime.
We shall prove that, given any solution of 2, the t(/,
yij appearing in #>, obtained from the solution, constitute
a solution of Q. This is obvious for the normal solutions
of (1). Then if O is a form in ,, (?, considered as
a differential form in the Ui, yj, holds 2'. This proves our
statement.
Now let (2) be a solution of 2 which annuls some St.
Consider the corresponding solution of Q. By 64, there
is a region 2T such that, given any *>0, we can find
a solution mk, yjk of ., with no Si zero at any point of 21',
such that (3) holds at every point a of SI'. We suppose 81'
to be taken so that the coefficients in (1) are analytic
throughout 21'.
Now if a is any point of 21', the tfc(a), yjk (a) in (3)
furnish initial conditions for a normal solution of the basic
set of differential forms (1) (65). Thus for any a in 21',
there exists a normal solution te,, yj of (1) which satisfies
(3) with the solution (2).
VI. CONSTITUTION OF MANIFOLDS 1Q3
We repeat the above procedure, using 2m and c/2 in place
of m and e. We find a region 21", in 21', every point a of
which can be used as above. Employing 4m and f/4, we
find a region 2f " in 21". We continue, determining a sequence
of regions 2I (0 . There is at least one point a common to
all of these regions. Given any >0, and any m, there is
a normal solution of (1), analytic at a, for which (3) holds.
As there is an a in every area in which (2) is analytic,
(2) is a seminormal solution of (1). Our result is proved.
It is very likely that the set of points a consists of all
points at which the functions in (2) are analytic, with the
possible exception of an isolated set. One might ask, also,
whether every seminormal solution can be approximated
uniformly in some area, with arbitrary closeness, by a normal
solution. A negative answer would certainly be interesting.
These questions need more attention than we have been
able to give them.
Example. Consider the form in the unknown ?/,
It is algebraically irreducible in the field of all constants
because, when equated to zero, it defines y s as a two-branched
function of y and y.
Equating A to zero, we find, for y 4 - 0,
dx y \ y
the solutions of which are given by
(6) y = b e ll ~*>
and
with b and c constants. The solution y = 0, suppressed
above, is included among these.
The solutions (6) with 6^0 are all normal. From the
fact that if b stays fixed in (6) at a value distinct from zero,
while c approaches GO through positive values, y approaches
104 ALGEBRAIC DIFFERENTIAL EQUATIONS
& uniformly in any bounded domain, we see that the solutions
y b with b ^ are seminormaL Consider the solution
y = 0. Let & have any fixed value distinct from zero. By
taking c as a sufficiently small negative number, we can make
the second member of (6) and an arbitrarily large number
of its derivatives small at pleasure at x = 0. This shows
that y = is a seminormal solution and that the general
solution of A is the whole manifold of A.
Of course, by taking 6 sufficiently small in (6), we can
approximate uniformly, with arbitrary closeness, to y = 0,
by means of normal solutions, in very arbitrary areas. But
the discussion above shows what might conceivably happen
in other examples.
74- We can extend the preceding results. Let F be any
form not in 2. It can be shown, precisely as in 13, that
if (2) is any solution of 2, there exists a set of points, dense
in S3, such that, given any point a of the set, any positive
integer m and any c >0, there exists a solution ti, ,%>,
analytic at a, for which F does not vanish and for which
(3) holds.
It follows that if O is a form with coefficients meromorphic
in 21, the coefficients not belonging necessarily to 3^, and if
G vanishes for every solution of 2 with F^Q, then O
vanishes for every solution of 2.
ADJUNCTION OF NEW FUNCTIONS TO of
75. Let 2 be a non- trivial closed irreducible system.
Assuming ^ not to consist purely of constants, we shall study
the circumstances under which 2 can become reducible through
the adjunction of new functions to $r, that is, through the
replacement of of by a field S\ of which o? is a proper subset.
The functions of $i are assumed meromorphic in SI.
We form a resolvent 1 for 2, relative to $, using a form
Pw Q as in 28. Let be the system of all forms in the
t, yi and w which vanish for all common solutions of 2 and
Pw Q for which P^ 0. Listing the unknowns in the order
VI. CONSTITUTION OF MANIFOLDS 1Q5
Mi, *, Uq\ W\ ?/i, , y p ,
we take a basic set
(7) A, Ai, , A p
for &, with A algebraically irreducible in S r . Then 4 =
is a resolvent for 2.
Suppose now that the irreducible factors of A in 3^ are
J8i, , jBs. Then each J?i is of the same order in w as 4.
For, let r represent the order of A in w. If the coefficients
of the powers of w r in A all had a common factor in e^,
they would have a common factor in cF, and 4 would be
reducible in S.
Consider the systems
(8) Bj, Ai, ., A p
/ = 1, -, s. Let / be the totality of forms in &i which
vanish for every solution of (8) which annuls no separant
in (8). Then / is irreducible in Si. Let 2j be the system
of those forms of 12/ which are free of w. Then, relative
to Si 9 2j is closed and irreducible.
We shall prove that 2 holds every 2j 9 that no 2 h holds
any 2k with fe ^ h, and that every solution of 2 is a solution
of some 2j. Thus, 2 i9 , 2 a will be a decomposition of 2
in Si , into essential irreducible systems.
Since
8 A _ T> I? ^i I
- - - Jj 2 . . . j$ s -- f- ..
every normal solution of A is a normal solution of some Bj.
Thus every normal solution of (7) is a solution of some /.
Hence a solution of 2 obtained by suppressing w in
a normal solution of (7) is a solution of some 2j. Every
solution of 2 with P 4 is obtained by a suppression of w
in some solution of 12.
Suppose now that some solution of 2 is not a solution of
any Sj. Let Cj be a form in 2j, j 1, - - ., s, which does
not vanish for the solution. Then Q C 8 does not vanish
for the solution. But Ci C 8 vanishes for every normal
106 ALGEBRAIC DIFFERENTIAL EQUATIONS
solution of (7). By 72, it holds . Hence it vanishes
for all solutions of 2 with P4 0. By 74, it holds 2.
Thus every solution of 2 is a solution of some 2).
If 8, the separant of A, were in some ty, it would be
divisible by JBj. Then A and S would have a common factor
in P!, hence, also, in ?, and A would be algebraically
reducible in S r .
Consider any form Tof . Any solution of (8), for some,/,
which annuls neither S nor any separant in (8) is a normal
solution of (7) and annuls T. Hence ST is in % so that T
is in /Jy. Then every form of 2 is in -/, so that 2; holds 2/.
Suppose that P is in some -0,. Then the remainder 1\
of P with respect to (7) is in - ( 2y. Then P, is divisible by fy,
and yi and I\ have a common factor in r . This is im-
possible, since Pj is of lower degree than A in ?r r . Then
P is not in any Q.
Now Pw Q is in every /. It follows easily that fy is
the totality of forms with coefficients in r t which vanish for
those common solutions of ~) and Pw Q for which P^O.
This means that if J2& held some 2\., where ft^^ ^hen
- r 4 would hold /4. Then Bh would be in /4 and would
be divisible by B k . Then A would have a double factor
in $1 and hence would be reducible in c?. Thus no 2^ can
hold a 2 fc with k 4-h.
Thus, /or ^ to be reducible relative to Si , it is necessary and
sufficient that the resolvent of 2 relative to $r be algebraically
reducible in ^ .
We see that Bj = is a resolvent for J/. Thus in the
decomposition of 2 into irreducible systems in i, every
essential irreducible system will have w t , -, u q as arbitrary
unknowns and the sum n -\ \-r p of 31 is the same for
all of the irreducible systems.
INDEGOMPOSABILITY AND IRREDUCIBILITY
76. Let 2 be an indecomposable system of simple forms
in 2/i > > y n the domain of rationality being a field r . We
VI. CONSTITUTION OF MANIFOLDS 107
shall prove that, if ~ is considered as a system of differential
forms, it is irreducible in *&.
We assume, as we may, that -2 is non-trivial. Let A be
the totality of simple forms which hold 2. Let (1) (with
the unknowns relettered) be a basic set for A. Let G and H
be differential forms such that GH holds 2. Let G\ and H L
be respectively the remainders of G and H with respect
to (1). Then G\H, holds 2.
We shall prove that one of 6ri, HI is identically zero.
Suppose that this is not so. Let GI and HI be arranged
as polynomials in the uij with j>Qj the coefficients being
simple forms in ?/!,--, y p . We understand that no coefficient
is identically zero. The coefficients, being reduced with respect
to (1), cannot hold -S. As 2 is indecomposable, there is
a regular solution of (1) which annuls no coefficient. Let
a be a value of x for which no coefficient, and no separant
or initial, vanishes. For x = a, and for the values of
i, - -, y p in the above solution at a, GI and HI become
polynomials y and h in the mj with j>l. Let numerical
values be assigned to these mj so that neither// nor h vanishes.
We now construct functions Wi,---,?^, analytic at a,
whose values at a are the values in the above solution and
whose derivatives appearing in G and H have, at a, the
values assigned to the u\j above. For these ut, (1) has a
regular solution in which all ie#, yi in GiHi have, at a, the
values used above. Then GiHi cannot vanish for this regular
solution of (1).
Thus, let G l vanish identically. Then G vanishes for all
regular solutions of (1). But every solution (2) of 2; can be
approximated uniformly, in some area, by a regular solution
of (1). The derivatives of the functions in (2) which appear
in G will be approximated by the corresponding derivatives
in the regular solution. Thus G vanishes for (2) and holds ^.
Then 2 is irreducible.
CHAPTER VII
ANALOGUE OF THE HILBERT-NETTO THEOREM
THEORETICAL DECOMPOSITION PROCESS
ANALOGUE OF HILBERT-NETTO THEOREM
77. In 1893, Hilbert, extending a result of Netto for
polynomials in two variables, proved the following remarkable
theorem. Let , -, a/-; 6, be polynomials in y, , y n
with numerical coefficients. Suppose that b vanishes for every
set of numerical values of y^, - - , y n for which a. , a r all
vanish. Then some power of b is a linear combination of
the aij with polynomials in y\ , , y n for coefficients*
The Hilbert-Netto theorem holds, with no modification of
the proof, for simple forms. If F if , F r \ G are simple
forms in y\ , , y n such that G holds the system F . , F rj
then some power of G is a linear combination of J?\, , F r ,
with simple forms for coefficients.
Assuming the foregoing result, we shall establish the
following
THEOREM. Let F ly , F r ; G be differential forms in
y\* ' > yni such that G holds the system JFi, , F r . Then
some power ofG is a linear combination of the F{ and a certain
number of their derivatives, with forms for coefficients.
78. The above theorem will be easy to prove, with the
help of an idea taken from Rabinowitsch's treatment of the
algebraic problem, after we have settled a special case.
*A very simple proof is given by A. Rabinowitsch, Mathematische
Annalen, vol. 102 (1929), p. 518. See also van der Waerden, Moderne
Algebra, vol. 2, p. 11 and Macaulay, Modular Systems, p. 48. Hilbert
gave results more general than the above.
108
VII. HILBERT-NETTO THEOREM 109
Suppose that the system F if , F r has no solutions. We
shall show that unity is a linear combination of the Fi and
their derivatives, with forms for coefficients.
We assume that unity has no such expression and force
a contradiction. First, we shall show that there exist n power
series
(1) coi + en (x a) + c 2 i (x a)*H ,
i = 1, , n, which, when substituted for the yt, render
each Fi zero. The series obtained may have zero radii of
convergence. The derivatives of the series are thus under-
stood to be obtained formally. In substituing the series for
the yi into a form, we use the Laurent expansions of the
coefficients in the form at a. After these formal solutions
are secured, we shall be able to show that there exist
solutions in which the yi are actually analytic functions.
79. We consider the system of forms consisting of
-Fi , , F r and of their derivatives of all orders, writing
the forms of the infinite system, in any order, in a sequence
(2) HI , J? 2 > j H g , - .
Similarly, we write all yy, arbitrarily ordered, in a sequence
(3) *i > 2 > > Zg ,
Each Hi will now be considered as a simple form in the Zi.
The domain of rationality will be r.
To show the existence of the formal solutions (1), it will
suffice to find a set of numerical values for the ^, and
a value of x at which the coefficients in the Fi are analytic,
which make every form in (2) zero.
Consider any non-vacuous finite system tf> of simple
forms Hi taken from (2). We shall consider the unknowns
in <Z> to be those which actually figure in the Hi in <Z>.
We know from the Hilbert-Netto theorem, as applied to
simple forms, that tf> has solutions. Otherwise unity would
be a linear combination of the Hi in <Z>, in contradiction of
the assumption in 78.
HO ALGEBRAIC DIFFERENTIAL EQUATIONS
Let q be any positive integer. We construct, in the
following manner, a system 2 q of simple forms in z 1? -, z q .
A simple form K in ^i, , z q is to belong to 2 g if there
exists a system tf>, as above, which K holds.* Every
2 q contains the form 0. When q is so large that an Hi exists
involving no q with j>q, 2 Q will have other forms than 0;
for instance, it will contain Hi.
We shall prove that, for every #, 2 q has solutions. We
need consider only the case in which 2 q has non-zero forms.
By 7, there exists a finite subsystem of 2 q ,
(4) K 19 ..-, Ks
which 2 q holds. With each Ki, there is associated
a system O t of forms (2) which Ki holds. The totality of
forms in <Z>! , , <Pj is a finite system A of forms (2).
Now A has solutions and each Ki holds A. Hence 2 q holds A,
and has solutions. Since every form in z ly , z q which
holds 2 q holds A, 2 q is simply closed.
Evidently, for every q, 2 q is contained in 2 ? +i and consists
of those forms in 2 q +i which are free of z q i.
80. For each g, let 2 q be decomposed into essential
prime systems
(5) //!, .--, lit.
Then 2 g consists of the forms common to all /T^.t
Let H' be any prime system in the decomposition (5) of 2 t .
We are going to show that there is a prime system 77 " in
the decomposition of 2 2 whose forms free of z% constitute //'.
Let
(6) AI , - - , A v
be the decomposition (5) of 2 t . Those forms of Ai which
are free of z 2 constitute a prime system ^. The forms
common to
(7) Vi,---,^
* <P may involve Zi not in K. Of course K need not be in (2).
t It is unnecessary to express, notationally, the dependence of (5) on q .
VII. HILBERT-NETTO THEOREM ]11
is the totality of forms in 2 2 which are free of ^ 2 , that is -\.
Then (7) is a decomposition of ^ into prime systems. For,
firstly, ^i holds each W im Again, if some solution of ^ were
not a solution of any ^, we could find a form Si in each Wi
which does not vanish for the solution. Then Si *S r ,
which is in every ^, hence in 2 1? would not vanish for
the solution.
Thus the decomposition (5) of -\ is formed from (7) by
suppressing certain ^-. Then some *Pi is identical with //'.
This means that there is some prime system //" in the de-
composition (5) of ~ 2 whose forms free of z constitute //' '.
Similarly, there is a prime system 11'" in the decomposition (5)
of 2 8 whose forms free of z% constitute //". We continue,
in this way, forming a sequence
(8) n',n",...,nw,....
81. We now form a system 12, putting into tt every form
which is contained in any of the systems (8). Any particular
form in 12 involves only a finite number of unknowns.
We are going to find a value a of x for which the co-
efficients in the Fi are analytic, and numerical values of
the Zi, for which every form in 12 with coefficients analytic
at a vanishes. Since every Hi of (2) is in - C 2, every Hi will
vanish for the values found, and we will have the formal
solutions (1).
There may be a Zi such that every form in 12 which involves
that Zi effectively, also involves some Zj with j ^ i. If such
Zi exist, we select that one of them whose subscript is a
minimum, and call it u t . It may be that there is some Zi
(not HI) such that no non-zero form of 12 involves only MI
and the new z%. If such Zi exist, we represent by u 2 that
one of them whose subscript is a minimum.
Continuing in this way, we form a set of unknowns m,
which is either vacuous, finite or countably infinite, such that
no non-zero form in & involves only the IH, while every zj
which is not a m appears in a non-zero form involving only
that Zj and the m.
112 ALGEBRAIC DIFFERENTIAL EQUATIONS
We order arbitrarily the Zi which are not among the Ui,
calling them v t , v 2 , etc. The sequence of vi, for all that we
can say offhand, may be finite or countably infinite. We
assume, in what follows, that the vi are infinite in number;
only trivial modifications of language are necessary when
their number is finite.
In using the terms "initial", "remainder", etc., below, we
shall understand that every m precedes every vj.
From among all non-zero forms in & which involve only
v\ and the ui, we select one, A lf whose degree in v t is a
minimum. There exist non-zero forms in which involve
only v$, Vi and the m, and which are reduced with respect
to AI . From among all such forms, we select one, A% whose
degree in v 2 is a minimum. Continuing, we form an infinite
sequence
(9) A lf A 2 , -, Aq,
We are going to show that it is possible to form (9) in
such a way that, for every q, the initial I q of A q involves
only the Ui.
This is true automatically for I t . Suppose, then, that we have
been able to arrange so that TI, , I q -\ involve only the m.
Let B be any non-zero form in , involving vi, -, v q
and the ?i,-, reduced with respect to AI, , A g -i, and of
as low a degree in v q as it can be with these conditions.
The system <Z> of all forms in which involve only
v l9 , v q -i and the m in B, A if , A q -i is a prime
system. This is because <Z> is contained in some 77U> and is
the system of all forms in that 77< which involve only the
stated Ut, v. Then At, , A q -\ is a basic set for <D. We
construct a simple resolvent E = for <2>, with
(10) W = Oi VH h flg-l Vg-l,
the a being integers.
The initial Q of B is not in , hence not in <Z>. When
we replace the vi in Q by their expressions in terms of w,
we get a relation
(ID
VII. HILBERT-NETTO THEOREM H3
where P and 8 involve w und the m, the relation holding,
where w is as in (10), for every solution of <Z> with $4
Then
(12) SQ P =
for every solution of <Z>, if w is as in (10). As Q is not in
<P, P is not divisible by R. Thus, we have an identity
(13) MP + NE = L
with L not zero, and free of w, that is, involving only the ^e^
From (12) and (13) we see that, for every solution of <Z>,
and for w as in (10),
(14) MSQ L = 0.
Then, if w is replaced in (14) by its expression (10), the
first member of (14) becomes a form in 0. We have thus
L = UQ + V,
with V in <Z> and U a form in the unknowns in <P.
Let B be of degree s in v q . Let
C =
Then C is in Q and is of degree s in v q , with L for initial.
The remainder D of (7, with respect to A i9 , -d^-i, will
be of degree s in v q . Its initial will involve only the ui.
We can use D for A 5 in (9). This proves our statement re-
lative to (9) and, in what follows, we assume that every I q
involves only the m.
82. We are going to attribute constant values to the Ui in
such a way that each Ii becomes a function of x which does
not vanish identically.
Each /,- has at most a finite number of factors of the
type Ui h, h constant. Thus the set of polynomials Wi h
114 ALGEBRAIC DIFFERENTIAL EQUATIONS
which are factors of one or more /* is finite or countable.
Then let ci be a constant such that no J z is divisible by
HI <v If we put ^ (\ in the I z , each It becomes
a polynomial J*, free of u { and not identically zero. Similarly
we replace n g in the J/ by a c 2 so that no J} vanishes
identically. Continuing, we replace all m by constants in
such a way thai each // becomes a non-zero function of ,r.
83. Let 33 be an area in 21 in which the coefficients in
the Fi are analytic. Then every H q in (2) has coefficients
analytic* in 33. The equation A l = 0, with the m fixed as
in 82, determines one or more functions vi, of x, analytic
in some area Si in 33. Let one of these functions be selected,
and substituted into A*. Then A 2 = gives one or more t- 2 ,
analytic in 33 2 contained in 33i. We substitute such a t- 2 ,
and the /; x selected above, into ^1 3 and solve A$ for v s ,
using an area 33 3 in 33 2 . We continue, finding a v q and
a 33g for every q. For any g, the functions ?i, , u q , to-
gether with the constant values attributed to the m in AI , , A q ,
annul those forms in which involve only the unknowns in
*TI , , Aq .
Let a be a point common to all areas 3v Then, a, the
values of the Vi at a and the constants selected for the M,
annul those forms in tt whose coefficients are analytic at a .
In particular, the Hi of (2), vanish for the above values.
This proves the existence of the formal solutions (1) of
(15) ^i,"-,-Pr.
84. We shall now prove that (15) has analytic solutions.
It is not difficult to see that the results of 7-14, and
also those of 23, 24 hold when a solution of 2 is defined
as any set of series (1) which formally annul every form in ~ .
With this new definition of solution, let (15) be decomposed
into closed essential irreducible systems 2 l9 , 2 8 . We
know of course from 78, that the Fi are not all zero. Let
?<i , , u q be a set of arbitrary unknowns for 2i and let
(16) <4i,---,4,
VII. HILBERT-NETTO THEOREM 115
each At introducing ?//, be a basic set for 2\. Let S\ be
the separant, and It the initial, of At.
We are going* to show that (16) has analytic regular
solutions.
Suppose that when the At are regarded as simple forms
in the yy, H } J which they involve, they have a solution,
consisting of analytic functions, which annuls no Si or /f.
Then the values of the u,j, yij at some suitable point will
furnish initial conditions for an analytic regular solution
of (16).
Now, if
T =-- Si S p TI J p
vanished for all solutions uy, y,j of (16) considered as a set
of simple forms, we would have, using the Hilbert-Netto
theorem as applied to simple forms, an identity
(17) T h - CVli-f..- +C P A P .
But (17) would continue to hold for all formal power series
solutions of (16) considered as a set of simple forms. Then
the basic set of differential forms (16) would have no regular
power series solutions.
This shows that (15) has analytic solutions. We have
reached a contradiction which proves that unity is a linear
combination of the Fi, and of a certain number of their
derivatives, with forms for coefficients.
85. We now complete the proof of the theorem stated in
77.*
We adjoin an unknown z to the y^ and consider the system
of forms
zG-\, F l9 , F r ,
which evidently has no solutions. Let K = zG \. Then
there exists an identity in the Zj, yg,
m j ; in r -t i
< 18 > '-<>
where the Cj, D,j are forms in z, y\, , y n .
' Cf. Rabinowitsch, loc. cit.
116 ALGEBRAIC DIFFERENTIAL EQUATIONS
If we replace z by 1/0 in (18) and each gj by the ;th
derivative of I/O, the first sum in (18) vanishes. We find
thus an identity
where the Ey are forms in
Then
and this establishes our theorem.
Example. We consider two forms in the unknown y,
F, = y\ F, = yi 1.
The system J5\, J^ has no solutions. We have
86. It follows from 84 that if a system of algebraic
differential equations in y\, , y n has formal power series
solutions j then the system also has analytic solutions.
One might ask whether a system 2 which is irreducible
when the solutions y^, , y n are understood to be analytic
functions, remains irreducible when the yi are allowed to be
formal power series. The answer is affirmative. Let OH
hold 2 according to the second definition. Then one of G,
H holds 2 for the first definition. Suppose that G does.
Let
(19) JSi, , B a
be a finite subsystem of 2 with the same manifold (first
definition) as 2. Then some power of G is a linear com-
bination of the Bt and their derivatives. Then G holds 2
for the second definition.
THEORETICAL PROCESS FOR DECOMPOSING A FINITE SYSTEM
OF FORMS INTO IRREDUCIBLE SYSTEMS
87. We deal with any finite system 2 of differential forms
in y l9 , y n . Let p be any positive integer. We denote
VII. HILBERT-NETTO THEOREM U7
by 2W the system obtained by adjoining to 2 the first
p derivatives of each of its forms.
When the forms in 21 & are regarded as simple forms in
the yij which they involve, 21& goes over into a system AW
of simple forms. For domain of rationality, we use of.
Using the method of 55-60, we decompose A ( &, by
a finite number of operations, into essential indecomposable
systems
(20) <!>!,... ,0> r .
Let the forms in the <P t be considered now as differential
forms in the y*. Then each <Z>< goes over into a system of
differential forms */ 7 t . Let any Wi which is held by some Wj
with j 4 i be suppressed. This can be accomplished by
a finite number of operations ( 68). There remain systems
(21) ^i,---, "V
We say that, for p sufficiently great, (21) is a decomposition
of 2 into essential irreducible systems.
88. Let
(22) 2i,---,2
be a decomposition of 2 into finite essential irreducible
systems. When the forms in the 2t are regarded as simple
forms in their y#, (22) goes over into a system of simple
forms
(23) /\, -,/.
Let us make any selection of t forms, one from each 2\-,
and take their product. Let the products, for all possible
selections, be
Aij A% 9 , Ag .
Then each At holds 2. By 77, if p is large, some power
of each At will be a linear combination of forms in 2W, with
forms for coefficients.
If then each Ai is considered as a simple form in its y#,
and if it is represented then by 2?<, each Bi will hold ^/ (p)
if p is sufficiently large. Let p be large enough for this.
118 ALGEBRAIC DIFFERENTIAL EQUATIONS
We shall prove that ^ach <&i of (20) is held by some /}
of (23).* Suppose that #>i is not so held. Let Cj be a form
of /}, j ^ 1 , , t which does not hold </>! . Then C\ CV,
that is, some J3,-, does not hold <t> { . Then that JB t - cannot
hold s/ (p} . This proves our statement.
It follows that each Wi is held by some 2/.
On the other hand, each ^ is held by some tyj. Let this
be false. Let Dj be a form in */{/, J = 1 , - , r, (we restore,
momentarily, the suppressed '/{/) which does not hold ^\ .
Then G - A 7>V does not hold ^ . Hence (r does not
hold 2. Then, if G is considered as a simple form in its yij,
it does not hold ^W. This contradicts the fact that ^ (p)
is equivalent to (20).
Thus, for p sufficiently great, (21) is a decomposition of 2
into essential irreducible systems.
For the above process to become a genuine method of
decomposition, it would be necessary to have a method for
determining permissible integers p.
This question requires further investigation. In 89, we
treat a special tfase.
Example 1. Let 2 be y\ 4//, in the unknown y. Then
st { & is equivalent to the system
y? *y, 2/iO/2 2), 2/12/3 + 2/2(2/2 2),
2/i 2/4 + 2 ?/ 2 2/3 + 2/3 (2/2 2), - -,
+ G> i) 2/p-i 2/3 -f 2/p (y ~ 2 )-
decomposes into the two indecomposable systems
(24) 2/,2A
in the unknowns 2/ ? 2/u 2/a- If we adjoin y 2/3+2/2 (2/2 2) to
(24), that system decomposes into
(26) y, yi, y*
(27) _ y> 2/i> ft 2, _
* The unknowns are all which appear in (20) and (23).
VII. HILBERT-NETTO THEOREM H9
which are systems in y, y lf y 2 , y$. The same adjunction
to (25) gives (27) and
(28) y\-ly, y, 2, y a .
Thus (26), (27) and (28) give the decomposition of A^\
Continuing, we find the decomposition of A^ to be, for
P>2,
(29) y, ?/i, </ 2 , , y p ,
(30) y, y lj s/2 2, y 8 , , y P ,
(31) y? 4y, 2/2 2, 2/3, , Jtefi.
If we regard the last three systems as systems of differential
forms, (31) gives the general solution of y\ 4y, while (29)
gives the solution y = 0, which is a second irreducible
manifold. The system (30) of differential forms has no so-
lution.
We notice that the system of simple forms y\ 4?/, y% 2
holds the system (30) of simple forms. This is in harmony
with the fact that every O* in (20) is held be some 7)
in (23).
Example 2. Let 2 be the form y\ 4y 8 , which, from the
fact that its manifold is y = \l(x a) 2 and y = 0, is seen
to be an irreducible system. If we let
and represent the rth derivative of AI by A r +i, then
will be
y\ 4y*, y^A v
VP A! + (p 1) yp-iA 9 H ----- h (p 1) 2/2 Ap-i + 1/! A p .
Then ^/ (1) decomposes into
(32) y, yi
(33) 2/?-42/ 8 , ^.
120 ALGEBRAIC DIFFERENTIAL EQUATIONS
We now examine ^/ (2) . The adjunction of y^Ai+y^A^ to
(32) gives the single system
(34) y, y l9 ys.
The same adjunction to (33) gives
(35) 0f-4y, A l9 A*,
and also (34).
Let us examine ^ (8) . The adjunction of y* AI + 2 1/ 2 A 2 -f ?/i A$
to (34) gives the single system
(36) y, ?A, y 2
in the unknowns y, , */ 4 . The same adjunction to (35)
gives
(37) y\-y\ A,, A,, A,,
as well as the system, held by (36), obtained by adjoining
y, to (36).
Continuing, it is not difficult to prove that the decompo-
sition of sl ( & is
(38) y, y l9 - , y q
where q is the greatest integer in 1 + p/2 , and
(39) yJ
The system (39) of differential forms gives the manifold
of 2, while (38) (differential forms), whose manifold is y = 0,
is held by (39).
FORMS IN ONE UNKNOWN, OF FIRST ORDER
89. Let A be a form in the single unknown y, of the first
order in y, and irreducible algebraically. We shall show
how to determine, in a finite number of steps, a finite system
of forms whose manifold is the general solution of A.
Let ^1 be of degree m in y l . We consider the system
(40) ^,4 1 ,...,^ m -i >
VII. HILBERT-NETTO THEOREM 121
there Aj is the t /th derivative of A. Let (40) be considered
as a system of simple forms, and let it be resolved into finite
essential indecomposable systems. There will be precisely
one indecomposable system, ^/, which is not held by S, the
separant of A ( 73). Let the forms of A be considered now
as differential forms in y. Let 2 be the system of differential
forms thus obtained.
We shall prove that the manifold of 2 is the general
solution of A.
90. We know that the general solution of A is contained
in the manifold of 2. What we have to show is that every
solution of 2 is in the general solution of A.
We observe that A holds 2. The solutions of A not in
the general solution are solutions of 8. The common solutions
of A and S are solutions of the resultant of A and S with
respect to y l9 which is a non-zero form jR, of order zero in y.
It suffices then to show that every solution u of R which is
a solution of 2 is contained in the general solution of A.
Let uj be the jth derivative of u. Then A for y = M,
yi = HI- There exists an open region 2li and an A>0 such
that, for
(41) x in Six and 0<|y u|<A,
every solution of the algebraic relation A = 0, for y^ con-
sidered as a function of y and x, is given by a series
(42) yi - ti! = ao(y tt)"+ + a p (y
where the at are functions of x analytic in 2ti and where
q and s are integers, s being positive. The particular series
used in the second member of (42) depends on the particular
solution yi used. But, for each such series, we have s <J m.
We suppose that, in each series, OQ does not vanish for every x.
The system of functions
(43) u, uu - -, u m
is a solution of A. By 64, there is a region 21 2 in 2d in
which we can approximate arbitrarily closely to (43) by a
9*
122 ALGEBRAIC DIFFERENTIAL EQUATIONS
solution of A with R distinct from zero throughout 21 2 . We
suppose 21 2 to be taken so that the coefficients in A are
analytic throughout 21 2 .
It follows that, if is any point in 21 2 , the differential
equation A - has solutions analytic at J, with R ^ at ?,
for which y, , y m differ arbitrarily slightly at from
?*, -, u m respectively.*
Any such solution satisfies (42), in the neighborhood of J ,
for an appropriate choice of the series in (42).t Hence
there must be one of the series for which (42) is satisfied
by a solution of A with JK 4 - and with y, -, y m as close
as one pleases at 5 to u, , u m . In what follows, we
deal with such a series and assume to taken so that
a,, 4 at .
We see first that # >0 in (42). Otherwise ?/i MI would
not be small at ? if y n is small at . Differentiating
(42) we find
(44)
Replacing ?/, H t in (44) by its expression in (42), we find
y* - MI - q , a\ (y - w) 8 "- 1 + &i (2/ - tt)^"-"- 1 + - ,
&
where the bi are analytic in 2li. We notice that, if m > : 2,
2q/s 1 >0. Otherwise y a u 2 could not be small at
when y u is small.
Similarly, we find
ym MWI
(45) = q _/2 ? _ ][ \ lmq__ m + 2 \ ( )m , s - m+l + . . . .
S \ 8 I \ 8 1
The coefficient in the first term of the second member is not
zero at ?. Hence the first exponent in the series in (45)
must be positive. That is,
Note that if R t at I for a solution of A, then S * at
H R at $, y u for a neighborhood of .
so that
Thus,
VII. HILBERT-NETTO THEOREM 123
.._+ 1>0>
S
mq i-i-^
- m + 1 ^
8 =- .9
^ 8,1
q>s \- -,
* mm
and, as s< w, we have #>s 1, so that q>s.
We are now able to show that u belongs to the general
solution of .4. In (42), we replace y u by v s . Then (42)
goes over into the differential equation
(46) s f-- = a Q t*-*H H h a p V+P- S + I H .
doc
Since the second member of (46) is analytic in v and x
for v small and x close to ?, then, if we fix z; as a small
quantity at , distinct from 0, (46) will have a solution
analytic at , not identically zero, and with any desired
finite number of derivatives as small as one pleases at .*
Then y u = V s , while not zero at , will be small at ,
together with as great a finite number of its derivatives as
one may choose to consider. Solutions of A, close to w, but
distinct from w, at 5, cannot make R = 0.
Thus, if u is a solution of S, as well as of jff, u is
a seminormal solution of A and belongs to the general solution
of A. If u is not a solution of S. y r^-tainly belongs to
the general solution.
* Equation (46) is satisfied by v = , and its solution is analytic in
the constant of integration.
CHAPTER VIII
ANALOGUE FOR FORM QUOTIENTS OF LtrROTH'S THEOREM
91. It is an important theorem of Liiroth that if a and ft
are rational functions of x, then a and ft are rational functions
of a third rational function, y, which, in turn, is a rational
combination of and ft*
We are going to prove the following analogue of Liiroth's
theorem.
THEOREM. Let a and ft be two form quotients ( 38) in
a single unknoivn y. Then there exists a form quotient y,
in y, such that
(a) and ft are rational combinations of y and of a certain
number of its derivatives,
(b) Y is a rational combination of et, ft and a certain number
of their derivatives.
The coefficients in the rational combinations are functions
of x in o?.
As to the degree of uniqueness of y, we prove that if Y\
and Y* are two possibilities for y, then y* = (#yi + &)/(cj'i4~^)
with a, by c, d functions of x in 3\
92. We prove the following lemma.
LEMMA. Let PI, - - -, P m , Q, R be forms in y, R not
identically zero. Suppose that the relations
Pi(y) - Pi(z)
~ ~
where y and z are analytic functions for neither of which
R vanishes, imply the relation
* Appel et Goursat, Fonctions Algtbriqucs, 2nd edition, vol. 1, p. 283.
Van der Waerden, Modem Algebra, vol. 1, p. 126.
124
VIII. LfiROTH'S THEOREM 125
Then the form quotient Q/H is a rational combination of
the Pi/Rj and of a certain number of their derivatives, mth
coefficients in \
The values 1 and 2 of m will suffice in our applications
of this lemma.
Let v^ ., v m \ w be new unknowns. Consider the forms
(3) vi-Pi(i= l,.-.,m); Rw Q.
As ig 32, the system 2 of all forms in the Vi, w 9 y which
vanish for all solutions of (3) with-K^O, is irreducible. It
is easy to prove that any one unknown in 2 is a set of
arbitrary unknowns.
We take vi as arbitrary unknown, and form a basic set
for 2
(4) A -.., A m , B, C
which introduces, in succession, v 2 , *,v m , w, y.
We are going to prove that B is of order zero in w, and,
indeed, that it is linear in w.
Suppose that B is of order greater than zero in w. Consider
any regular solution of (4) with R(y) ^ 0. Such solutions
exist because R(y) is not in 2. Without disturbing the Vi
in the solution, we can alter the initial conditions for w
slightly at some point and get a second regular solution
with R(y) ^ 0. This would be contrary to the hypothesis
of the lemma.
We shall prove now that B is linear in w. Suppose that
this is not so.
Let AS, -, A m be of the respective orders r t , , r m in
v 2 , - , v m . Let C be of order r in y.
In (4), we replace the symbols
* w >
by
126 ALGEBRAIC DIFFERENTIAL EQUATIONS
respectively. The remaining v# and \ji we replace, in any
order, by symbols Ui (see 65).
Then (4) goes over into a basic set
(5) F*,...,F m , D, E
of a prime system.
Let
(6) CV--,^
be analytic functions of the m and x which annul F 2 , , F m
when substituted for z t , , z m ( 45). As D is of degree
at least 2 in s m +i, we can get two distinct functions,
Ci" 1 "^ and dr +1) which, with (6), annul D ( 46). After
treating E we will have two sets of functions
,,, i
^'' .// v(m) v(m-f-l) v(w+2)
b , ' ' ', > , fe2 , fe2
which annul (5) when substituted for the Zi. No separant
or initial in (5) is anulled by either set (7).
Let T be the remainder of R with respect to (4). Let
new Ui be taken, to correspond to the vu in T which are
not in (4). Let T go over into a simple form ?7in the %, Zi.
As U is reduced with respect to (5), it will not be annulled
by either set (7).
We attribute numerical values to x and the m in the
following way. We require the functions (7) and the coeffi-
cients in (5) and in U to be analytic for these values. We
require, secondly, that $"+ ^^ 4- 0. Finally, we ask
that U and the separants in (5) do not vanish for either
set (7), for these values.
The values chosen furnish initial conditions for two solutions
of (4) which annul neither R nor any separant. We use the
same vi in both solutions. Then t' 2 ,---,t' m are the same
for both solutions. On the other hand, w will not be the
same in both solutions. This contradicts the hypothesis of
the lemma.
VIII. LUROTH'S THEOREM 127
Then B is linear in w. If we replace n in B by PJR
and w by Q/ R, the resulting expression in // must vanish
identically. This completes the proof of the lemma.
93- Let A(y) and B(y) be two non-zero forms in //,
relatively prime as polynomials in the iji and not both free
of y. Let r be the maximum of their orders in //. We
shall prove that
(8) A(y)B(g)-B(y)A(g)
is not divisible by any form, not a function of x, which does
not effectively involve y r .
Suppose that there is such a factor, C, free of y r . Let
A(y) = J/^+.-.+.Vfc, B(y) = #<,*+ "+ A *>
where it is possible that either 3/ or N is zero. Then C
must be a factor, for every i, of
We shall show that C cannot be a form in z alone. Suppose
that C involves only z. Since A(y)IB(y) is not a function
of Xj we can assign two distinct sets of rational numerical
values to y, , y r in such a way that
AI B 2 BI A 2 ,
where the subscripts correspond to the substitutions, is not zero.
As, by (8)
At B (z) B, A (z} , A*B (z) B 2 A (z)
are divisible by C, then B(z) and A(z) must both be divisible
by C. Thus C would have to be a function of x.
Let C be of order s ;> in y. Let = A(z)/B(z)
and let
Wi = Mi aNi, i = 0, , h.
For any rational numerical values of g, -, g r for which
B(z) ^ 0, and for which the coefficient of the highest power
128 ALGEBRAIC DIFFERENTIAL EQUATIONS
of y 8 in C does not vanish, the expressions wi will all have
a common factor, which will be a polynomial in the yi, with
coefficients in S r .
Let
\~VhWhj
where the iii, Vi are indeterminates. Then, for arbitrary
rational Ui and vt, and for rational Zi as above, w u und w v
will both be divisible by a polynomial effectively involving y 8 .
Then the resultant Q of w u and iv v with respect to y s must
vanish identically in the m, Vi and x, y, , y r -i with i/ 5
omitted, if is obtained by the indicated substitutions for the Zi.
Now Q is a polynomial in . Since a depends effectively
on the Zi, we can find an infinite system of sets of numerical
values for the ^, as described above, each set giving a distinct
result for . Thus Q is identically zero, even in .
Then w u and w v have a common factor which is a poly-
nomial in the M,-, tv, ?A and , with coefficients in S r . Thus,
the expressions wt, with a indeterminate, must all have
a common factor d, which is a polynomial in y, , y r -\ y
a with coefficients in r .
If 8 were free of a, every Jlf t and every Ni would be
divisible by 6. Then A and B would not be relatively prime.
Thus d is of the first degree in .
Then, for every i, we have
with Hi, E, F forms in t/. Then (8) has a factor
which is free of z. This is impossible, for the same reason
for which C, above, could not be a form in z alone. The
proof is completed.
94. We proceed with the proof of the theorem stated in
91. When a and ft are both free of y, we take Y = 1.
In what follows, we assume that and ft are not both free
of y. We may write and ft with a common denominator.
Let a = P/R, ft = Q/R. Consider the forms in y and z
VIII. LUROTH'S THEOREM 129
P(y)R(z)-P(z)R(y),
which are not both identically zero.
Let 2 ly , 2 S be a decomposition of (9) into closed
essential irreducible systems. From each 2 if we select
a non-zero Oi which is of as low a rank as possible in z.
We assume, as we may, that each Oi is algebraically irre-
ducible. There must be some Oi which involves both y and
z effectively. Otherwise, (9) would imply a relation of the
type
C(y)D(e) = 0.
For z = y, this would become C(y) D(y) = 0. But (9) is
satisfied for z = y with y arbitrary.
Let GI, - , Op involve both y and z, while G p +i, -, G 9
involve either y alone or z alone. For i<p, the manifold
of 2i is the general solution of Oi.
Let Gp+i, - - , O q involve only z, and G q +i, , 8 involve
only y. Let
M= Gp+i'-.Gg.
Then M is not in any 2i, i < p, because, in the general
solution of Oi, i <J p, z can be taken almost arbitrarily.
That is, if we take the unknowns in Oi in the order 2, y,
then, given any regular solution of (?,-, we can modify z and
any number of its derivatives, at some point, slightly, but
otherwise arbitrarily, and get a second regular solution of Oi.
Let the greatest of the orders of G if , O p in z be r.
Let Oi be of order r in z f or i <[ m , and of order less than r
for m<i <i p .
Let
(10) H=Gt...G m .
We write J5T as a polynomial in z r . Let
(11) H = JV*-
There must be some ratio FJF which is not independent
of y. Otherwise some factor of F would be a factor of
130 ALGEBRAIC DIFFERENTIAL EQUATIONS
every Fi and // would have an irreducible factor not in-
volving z r . This would contradict (10). Let Ft/F be not
independent of y.
Let K = 9 Hjdzr- Because GI, , G m are .algebraically
irreducible forms, and none of them divisible by any other,
the resultant U of H and K with respect to z r is not identic-
ally zero.
Understanding that the unknowns in H have the order y, z y
let B be the remainder of M with respect to H. We say
that B and H are relatively prime polynomials. If, for
instance, B were divisible by (?i, some K F h M would
hold 2i. Now F is not in 2 l9 since it is of lower rank
than GI in z. Again,
(12) K = -^ 0, . . . m + - + 4,- Gi 0m-i.
d 2 r 9 r
Each term after the first in the second member of (12) is
in 2i . The first term is not. Thus K is not in ~\ . As M
is not in 2 lf # and are relatively prime.
Thus the resultant V of H and jB with respect to z r is
not identically zero.
Consider the form
S = UVFG m +i 0,.
We can assign rational numerical values to 0, , 2>_i
so that S becomes a non-zero form T in ?/, and so that
_F/F becomes a form quotient y in y which is not a function
of x.
We say that y as thus determined satisfies the conditions
of 91.
95- Let
(13) L(y) = TR(y)G q+l ... G 9 .
Let ?/ and y be two functions of x. analytic in some part
of SI, for neither of which L vanishes, and for which
R(y)
VIII. LUROTH'S THEOREM 131
Functions ~y ", y exist. For instance, we can use any // such
that L(y) $- and then take y = y. Let a be any value
of x for which ?/, // and the coefficients in R and every
G t , i = 1, , s are analytic, and for which L(yf) L(y)]-0.
By the definition of L(y), the equation H = 0, where
5-, , 2V-! are replaced by the rational values used above,
x by a and ?/ and its derivatives by the corresponding values
for y at a, will determine /* distinct values of z r .* To each
such z r will correspond a regular solution,t with y ~ y, of
some 6rj, ? <| m, which is not a solution of any Gi with
? >m. (See (12)).
A similar result holds for y.
Now, for y equal to y or to y, (9) will have the same
solutions in z. Let z, , z r \ have, at a, the rational
values used above. With these initial conditions for z, no
solution (y, z) or (y, 0) of (9) will be a solution of a JSi
with m < ? < j9 or q < ?' <I 6?. There will be 7* solutions
(y, z) and A solutions (y, z), which annul the form //. None
of these 2h solutions will annul M(z) and they will be the
only solutions of (9) with y = y or y and with z, , ^ r _i
as indicated at a, for which Jf(#) 4 0. Thus, in the h so-
lutions (y , 2-) which annul 7/, the A functions z must be the
same as in the (y, z) which annul //.
Then, for y = y or for y ^ ?/, the numerical equation
7f = for z r must have the same h roots for z r . We may
let the value a, of x 7 range over an area. Thus, in the ex-
pression for H in (11), every ratio Fi/F, with z, , z r -\
rational as above, must be, for every x, the same for y as
for y.
Consider then /. For
- R(u)L(~y)
and for ft(y) fl(y), similarly, we have
r(y) =
* Consider that S is divisible by F U.
fFor the order y, z.
132 ALGEBRAIC DIFFERENTIAL EQUATIONS
By 92, Y is a rational combination of , ft and a certain
number of their derivatives with coefficients in \
96. We prove now that a and ft are rational in Y and
its derivatives.
In Ft and F, with z, , z r -\ rational as above, let y be re-
placed by z. There will result two forms, At(z) and A(z). We wish
to show that neither At nor A has a higher rank in z than H.
Evidently, it is enough to show that if / is the form ob-
tained from H by interchanging y and z, then I is not of
higher rank than H in z.
For i^p, let Ei be the form which results from Oi when
y and z are interchanged. Then EI, -, E p must be mul-
tiples of 0^ , , Op taken in some order. For, since (9)
is symmetrical in y and z, the interchange of y arid z in
^ , , 2 3 will accomplish an interchange in pairs of those
systems. No 2i with i <p can be converted into a 2j with
j>p. Otherwise 2* would contain a form in y alone or in
z alone. This is impossible, for as was seen above, either
of y and z can be taken almost arbitrarily in the general
solution of Oi with i<Lp.
Then 2\ , - , S p are permuted among themselves. If ^
and Jfy are interchanged, then Ei must hold 2). Then JS*
is not of lower order than Oj either in z or in y. By sym-
metry, J?i and Oj have the same order in z. Then J2i must
be divisible by Oj, so that, as Ei and Oj are algebraically
irreducible, their ratio is a function of x in r.
As (?t has a greater rank in z than Oj \ii<m and j > m,
it follows that I is not of higher rank in z than H.
Let (7 be a highest common factor for At and 4. Let
A t = CW t and A = CW with F< and F relatively prime.
We are going to show that
which is not zero, equals H multiplied by a function of x.
We observe first that J is not of higher rank than Hin z.
Taking the unknowns in H in the order y, z, we let Y
be the remainder of L(z\ (see (13)), with respect to H.
VIII. LUROTH'S THEOREM 133
Then Y and H are relative prime. Let Z be the resultant
of H and F with respect to z r . Then Z is not zero.
Let real or complex numerical values be assigned to
,-, z r -i in such a way that UZF does not vanish iden-
tically in y and its derivatives. Let
Zi(y) = L(y)UZF,
where the substitutions just indicated have been made in
U, Zj F. We observe that the coefficients in LI may not
be in S r .
Then, if y is an analytic function for which Li(y)^Q,
and if a is a suitably chosen value of x, for which LI (y) ^ 0,
the differential equation H = 0, with z, -, r -i as just
taken, at a, will determine h distinct functions z for which
L(z)^Q and for each of which one has
(*) - (</); (*) - /%)
Hence, for the chosen y and for each such 0, one has ,7"= 0.
If we modify the numerical values attributed to z, , 2>-i
slightly, but arbitrarily, and make arbitrarily slight variations
in the values of y, , y r at a, leaving the higher derivatives
of y alone, we will still have L^ (y) ^ and we will get h
new functions z, which, with the new y, will make J zero.
All in all, we see that in some open region in the space
of x, y, , y r \ z, - , ? r i, the equation J = for z r ad-
mits all of the roots of the equation H = for z r . Then,
as H has no repeated factors, J is divisible by H. As J
is not of higher rank than H in z and as J has no factors
(not functions in S r ) which do not involve z r ( 93), we have
J= fiHy with fi a function of x in S r .
Let N be the remainder for R(z) with respect to Jff for
the order y, z. The resultant X ot H and JV with respect
to z r is not identically zero. Let rational values be substituted
for z, , z r -i in X and in FU of 94 so that XFU does
not vanish. Let D be the non-zero form in y which R(y) XFU
becomes for these substitutions.
134 ALGEBRAIC DIFFERENTIAL EQUATIONS
Now, let y and y be functions for which
(14) W(y) D(y) W(y) D(y) + 0,
and which, when substituted for y and z in J, render Jzero.
Let a be a value of x for which (14) does not vanish. The
differential equation H = for z y with y = y, has h solutions
z with ,--, 2V- 1 assuming the above rational values at a,
and with jK(?) ^ 0. The A pairs of functions (//, z) thus
obtained are solutions of (9). Similarly, we get li pairs (y, z)
which are solutions of H and of (9). It is easy to see,
because #= J/p, that the functions z in the li pairs (y, z}
are the same as those in the (?/, z). It follows that
Then and are rational combinations of Wt(y)IW(y)
and its derivatives. As y Wt(y)/W(y), the theorem of
91 is proved.
The above proof, and the methods of Chapter V, contain
everything essential for the construction of y in a finite
number of steps.
97. Suppose that we have two form quotients like y above,
Yi and ft. We see immediately that ft is rational in terms
of y l and its derivatives and that y is similarly expressible
in ft.
We now apply the method of 92. Let v = y l9 v 2 = ft.
We see that the system 2 of all forms in v^ v* which vanish
identically in y for i\ = y , v 2 = ft is irreducible. Let A
be a non-zero form of 2, of a minimum rank in t? 2 . Ac-
cording to 92 (YI can be made to correspond to P^/R
and ft to Q/R)j A is of zero order in v 2 , and is linear in t; 2 .
Similarly if B is a non-zero form of 2 of a minimum rank
in vi , then B is linear in v i . If we take A and B algebraically
irreducible, as we may, each will be divisible by the other.
Hence A is linear both in v and in v a . This proves that
Yi and ft are linear fractional combinations of each other.
CHAPTER IX
RIQUIER'S EXISTENCE THEOREM FOR ORTHONOMIC SYSTEMS
98. In Chapter X, we shall extend some of the main results
of the preceding chapters to systems of algebraic partial
differential equations. We shall find it necessary to use an
important existence theorem due to Riquier. We develop
this existence theorem now, following, in some respects, the
concise exposition of Riquier' s work given by J. M. Thomas.*
For the proof, in Chapter X, that every system is equivalent
to a finite number of irreducible systems, only 106 of the
present chapter, which can be read immediately, is necessary.
MONOMIALS
99. We deal with ra independent variables, x\ , , x m .
By a monomial, is meant an expression x\ l x l , where
the ik are non-negative integers. If a = yft, with a, ft, y
monomials, then a is called a multiple of ft. Given two distinct
monomials,
/y.*l . . . ~> l m rfJ\ . . . /v*^m
*! ^m > *! ^m >
the first is said to be higher or lower than the second according
as the first non-zero difference 4 jk is positive or is negative.
The following theorem, due to Riquier, is used only in
Chapter X.
THEOREM: Let
(1) i, 2, ? ?>
be an infinite sequence of monomials. Then there is an
which is a multiple of some aj with j <i.
* Annals of Mathematics, vol.30 (1929), p. 285. References will be
found in this paper.
135 10
136 ALGEBRAIC DIFFERENTIAL EQUATIONS
Let A be one of those t for which the exponent of Xi
is a minimum. Consider the monomials which come after fa
in (1). Let fa be a monomial of this class whose degree
in Xi does not exceed that of any other monomial of the class.
Of the monomials which follow fa , let fa be one of minimum
degree in x lt Continuing, we form an infinite sequence of
monomials
(2) A, A, A, -
whose degrees in a* are non-decreasing. We extract similarly,
from (2), a sequence in which the degrees in x 2 do not
decrease. We arrive finally at an infinite subsequence of (1)
in which each monomial is a multiple of all which precede it.
DISSECTION OF A TAYLOR SERIES
100. Let
be the Taylor expansion at
(4) Xi = 0, i = 1, , m
of a function u of x if , x m analytic at the point (4).
Let [] be any given finite and non-vacuous set of distinct
monomials. We are going to separate (3), with respect to [],
into a set of components.
Let a be the greatest exponent of x v in the set []. We
write
(5) u
where, fori<a, x\f i contains all terms in (3) in which the
exponent of x l is precisely i. As to af/ a , it contains all
terms divisible by o^. Then f v , f a _ 1 are series in
#2, , Xm, while f a involves also XL*
We define sets of monomials [a]*, A = 0, -, a, as follows.
If [] contains monomials in which the exponent of XL does
* We consider every combination ti, , im to occur in (3), using zero
coefficients if necessary.
IX. RIQUIER'S EXISTENCE THEOREM 137
not exceed A, then []A is to consist of all such monomials
in []. If there are no such monomials, then [a]^ is to be
unity. Let [0]*. be the set of monomials in # 2 , -, x m obtained
by putting x^ = 1 in [a]*. We now give to each /A, with
respect to # 2 , the treatment accorded to u , above, with respect
to #1. For A<a, we get a representation of the type
where b depends upon A, the/ju with i<.b involving x s ,--,x m ,
while /A6 involves also # 2 . For A = a, each / at - involves o?i.
That is, in the dissection of / a , we treat ^ like x 8 , -, x m .
We now operate on each f^ with respect to x s . We use
a set of monomials fy]^, where, if f/8]j, has monomials of
degree not exceeding ft in x%, [y]^ is obtained by putting
#2 = 1 in all such monomials, and where, otherwise, [y]^
is unity.
Continuing, we find an expression for u,
the summation extending over a finite number of terms.
Example: Let u be a function of x, y, z. Let [] be
xz*j xy, x*yz.
For Xj we find
w = /o (y, ) + a?/i to, ^) + ^ 2 /2 (xj y, z).
We now treat each/* with respect to y, the set of monomials
being that indicated below;
y,z g, y,yz.
Hence
to, ^) ,
(x, y, z) == /, (x, z) + y/ M (a?, y , *) .
10*
138 ALGEBRAIC DIFFERENTIAL EQUATIONS
The final step is
/oo(2/, z) =/ooo(y, *) 1;
/10 (*) = /100 + Z/101 + * 2 /102 (*) * 2 ;
/a (y, z) = Aw(y) + */m (y) + *Vii2 (y, *) 1,
/2o (# , s) = /2oo (#) + 2/201 (a?) + 2 /202 (a; , z) z*\
fu(x,y,z) = f2io(x,y)+zf2ii(x,y) J rz 2 f2i2(x,y,z) l,
Thus the dissection of u is
w = /ooo (y , *) + ^/loo + #/ioi + #* 2 /io2 0?)
+ #y/no (y) + xyzfm (y) + xyz*fn* (y, z)
+ ^/aoo 0*0 + ^ 2 ^/2oi (a?) + # 2 ^ 2 /202 (, ^)
101. Consider any monomial a = o x^ 1 in [a] and
any monomial ft in the expansion of u which is a multiple
of . Of course, ft appears in one and in only one of the
terms in the second member of (7). Let it app'ear in
X l l ' ' ' X m fi -i ' ^ 6 S ^ a ^ P r0ve ^ at x l l ' ' ' x m ^ S a mU ^"
1 "
tiple of a . For m = 1 , this result certainly holds. Let
the result be true for m = r 1 . We shall prove it for
m = r. We observe first that in the resolution (5) of w,
ft appears in a term x[ l f t with i^j r
Suppose first that ii < a in (5). Then ft/x[ l is free of XL .
Among the monomials used in the dissection of f^ will be
#2* x j r r and ft/x^ 1 will be a multiple of ocfy x j r r . As
there are only r 1 variables involved now, ft/x[ l will appear
in a term cfi^.. >i r in the dissection (7) of /^ with divisible
by #2* x j r r . Thus x[ l x l r r is divisible by .
Suppose now that ^ = a. Then //# is contained in / a .
Among the monomials used in the dissection of f a will be
x i* ' ' ' x i r * Now the formal scheme in (7) of the dissection
of f a can be obtained by taking a function g of x 2 , , XT,
dissecting g with respect to the monomials associated with / a
and then adjoining a?i to the variables in the series yielded
by g. That is, the monomials #** x l r r in the dissections,
IX. RIQUIER'S EXISTENCE THEOREM 139
analogous to (7), of / a and g, will be the same. Let y result
from ft on putting x^ = I . Then y is found in the dissection
of g with an x x l r r divisible by xi* - x j r r . The same
would therefore be true for ft/x a in the dissection of / a .
This completes the proof.
It follows that every monomial in [] is an #* 1 x% in (7).
1 02. The set of monomials consisting of all a?* 1 x% in (7)
which are multiples of monomials in [a] will be called the
extended set arising from [a] . The set of monomials a?* 1 x%
in (7) not in the extended set will be called the set com-
plementary to [].
If [a] is identical with the extended set arising from [],
then [] will be called complete.
Consider a set [a] which is not complete. We shall prove
that it is possible to form a complete set by adjoining to [a]
multiples of monomials in [].
Let p be the maximum of all exponents in all monomials
in []. Then, in (7), no 4 exceeds p.
Let []' be the extended set arising from []. Then if
[]' is not complete, it is a proper subset of its extended
set [a]" ( 101). Since we can never get more than (p + l) m
monomials x[ l x% in (7), this process of taking extended
sets must bring us eventually to a complete set.
103. In (7), the variables in an /^...^ will be called the
multipliers of the corresponding x[ l x%, and all other
variables will be called non-multipliers of x[ l x%. Of
course, if /^...^ is a constant, x[ l x% has no multipliers.
Let ft = a?* 1 a?!* be a monomial in the extended set
1 Tfl
arising from [a]. Let xt be a non-multiplier of ft. Then
/tftf/c, as a multiple of some monomial in [], is the product
of a monomial y in the extended set by multipliers of y ( 101).
We shall prove that y is higher than ft. Let y = a^ a.
If .71 < *'i , Xi cannot be a multiplier for y since ;'i is certainly
not the maximum of the degrees in xi of the monomials in [].
Hence j l > %. It remains to examine the case in which
ji = ii. When we dissect /^, we find that if jt< i 2 , x 2 can-
140 ALGEBRAIC DIFFERENTIAL EQUATIONS
not be a multiplier for xt* - ofy. Hence fa > i* and we
have to study the case in which j* = i* . Continuing, we
see that y is not lower than ft so that, since y ^ ft, y is
higher than ft.
104. We associate with every monomial x^ xfa the
differential operator
Then the product of two operators corresponds to the product
of the corresponding monomials.
Consider any monomial ft = x[ l x% in (7). Let the
corresponding differentiation be performed upon u, and after
the differentiation, let the non-multipliers of ft be given zero
values. Every term in the expansion of u which is not
divisible by ft will disappear during the differentiation. Any
term divisible by ft whose quotient by ft contains non-
multipliers of ft will disappear when the non-multipliers are
made zero. Hence the above operation gives identical results
when applied to u and to ftf^.-.i^
105. We study, for its instructive value, rather than
for purposes of application, a special system of equations.
There will be as many equations as there are monomials
x l l ' " x m* n (') (A set l] is supposed to be given.) With
each ft = x[ l - x% in (7), we associate an equation
where ^...^ is an arbitrarily assigned function of the
multipliers of /, analytic for small values of the multipliers.*
We shall show that there is one and only one function u,
analytic for Xi = 0, i = 1, - , m, such that each equation
(8) is satisfied, when the non-multipliers of the associated ft
are zero, for small values of the multipliers. We are not
finding an actual solution of the system (8). We are finding
* If ft = 1, the first member of (8) is u.
IX. RIQUIER'S EXISTENCE THEOREM 141
a u which satisfies each equation on a spread associated
with that equation.
Consider any particular equation (8). Let the second
member be integrated i times in succession with respect
to #1 from to Xi , then 4 times in succession with respect
to # 2 from to x* , and so on. The result will be a function
(9) x il x im 4
^ ' ^l ^m /V*m
where /^...^ is a function of the multipliers of ft, analytic
for small values of its variables. The function (9) satisfies
identically its associated equation in (8). Let u be the sum
of all functions (9) obtained from (8). The expression for
u as a sum gives the dissection (7) of u relative to [a].
Then, by 104, u satisfies each equation (8) on the spread
associated with that equation. If u is a second solution of
the problem, it is seen, from 104, that in the dissection
of HI u relative to [], the series fi^.-i m are all zero.
Hence u is unique.
MARKS
1 06. Let j/i, , y n be analytic functions of x i9 , x m .
Biquier effects an ordering of the yt and their partial
derivatives in the following way.
Let s be any positive integer. We associate with each
Xi any ordered set of s non-negative integers
(10) Mil, , His
in which the first integer, UH, is unity. With each y*, we
associate any ordered set of non-negative integers
(11) vu, -, t?i,
taking care that y { and j/j with i ^ j do not have identical
sets (11). The ^th integer in (10) is called the ^'th mark of xt,
and the jth integer in (11), the ^'th mark of y t .
If
a fc t n \-k m
(12) w = ^ s-y,
'
142 ALGEBRAIC DIFFERENTIAL EQUATIONS
we define the Jth mark of w, j 1 , , s to be v
~T" -j- Km Umj
Consider all of the derivatives of all j/ r * Let w l and
be any two of these derivatives. Let the marks of
and w 2 be
respectively. Suppose that the two sets of marks are not
identical. We shall say that wi is higher than w 2 or is
lower than w 2 according as the first non-zero difference en bi
is positive or is negative. If the two sets of marks are
identical, no relation of order is established between u\ and w 2 .
If wi is higher than tv%, dwi/dxi is higher than dw*/dxi.
Also, dw/dxi is always higher than w.
When the marks in (10) and (11) are such that a difference
in order exists between any two distinct derivatives, the
derivatives of the y i are said to be completely ordered.
Suppose that the ordering is not complete. We shall show
how to adjoin new marks, after Ui S and Vis, so as to effect
a complete ordering. Clearly, the adjunction of such new
marks will not disturb any order relationships which may
already exist.
Let m additional marks be assigned, as in the following
table:
x i X 2 x m 2/1 2/2 y n
s+l 1 ... 0, 00 ... 0,
s + 2 1 ... 0, 00 ... 0,
s + m .. 1, 00 ... 0.
Now, let wi and w a be two derivatives with the same set
of s + m marks. The (s + i)th mark of wi or w 2 , i = 1 , , m,
is the number of differentiations with respect to xt in wi
or w*. Hence the same differentiations are effected in w^
as in w*. From the definition of the marks of IV L and ie/- 2 ,
it follows now that the functions of which w and w% are
* Each y. wiU be considered as a derivative of zero order of itself.
IX. RIQUIER'S EXISTENCE THEOREM
derivatives have the same sets (11). Thus n\ and w 2 are
identical, so that the new ordering is complete.
In everything which follows, we shall deal only with
complete orderings.
107. Let t , , m ; 1 , , n be variables. We associate
with w, in (12) the monomial j l ?^* r
Let ivi , , w;* be any finite number of distinct derivatives
of the y t . Let the monomial associated above with iv it
i = 1, ...,<, be a*. Let g be any positive number. We
shall show how to assign, to the $, &, real values, not less
than unity, in such a way that, if wi is higher than Wj, we
have, for the assigned values, *>#/.
We introduce s new variables z, , 5 - With each &
we associate the monomial z^ - - z u s " where the w.. are the
marks of #.. With each f . we associate /" /'' where
* t i o
the v f/ . are the marks of y r Then each f goes over into
a monomial ft i = z{ l - - z a s * with aj the jth mark of w..
It will evidently suffice to prove that we can attribute to
the Zi real values not less than unity in such a way that
fti>gfy if Wi is higher than Wj.
*Let r be the maximum of the degrees (total) of the A.
Let k be any positive number, greater than unity and greater
than g. We put
(13) Zi =
Then, if
with an > &/i, we have, for (13),
fti \ Zh -^
ORTHONOMIC SYSTEMS
108. Let 2/1, , j/n be unknown functions otxi, , a?m,
whose derivatives have been completely ordered by marks.
We consider a finite system <r of differential equations,
144 ALGEBRAIC DIFFERENTIAL EQUATIONS
84 ...8. --
A m
where
(a) in each equation, g is a function of Xi, , x m and of
a certain number of derivatives of the yi y every derivative
in g being lower than the first member of the equation;
(b) the first members of any two equations are distinct;
(c) if w is a first member of some equation, no derivative of
w appears in the second- member of any equation;
(d) the functions g are all analytic at some point in the space
of the arguments involved in all of them.*
We do not assume that every yi appears in a first member.
Riquier calls such a system of equations orthonomic.
The derivatives of the yi which are derivatives of first
members in the orthonomic system are called principal de-
rivatives. All other derivatives are called parametric deri-
vatives.
109. Given an orthonomic system, a, we shall show how
to obtain an orthonomic system with the same solutions, in
which, for each yi appearing in the first members, the mo-
nomials corresponding to those first members which are de-
rivatives of yi form a complete set ( 102).
Let equations be adjoined to (14), by differentiating the
equations in (14), so that, for each yt which occurs in some
first member, the monomials corresponding to the enlarged
set of first members constitute a complete set. By 102,
this can be done. We obtain thus a system cr, of equations.
Certain first members in 0i may be obtainable from more
than one of the first members in cr. In that case, we use
any one of the first members in <x which is available.
Consider any one of the equations in <r. Let w represent
its first member, and v the highest derivative in the second
member. If we differentiate the equation with respect to
ad, the first member becomes div/dx*. The highest deri-
* Thus, in (d), derivatives not effectively present in a g may be re-
garded as arguments in that g . This does not conflict with (a), in which
the arguments considered are supposed to be effectively present.
IX. RIQUIER'S EXISTENCE THEOREM 145
vative in the new second member will be dv/dxi, which is
lower than dw/dx t ( 106).
It is clear on this basis, that oi satisfies condition (a).
We attend now to (c). Let ( be an open region in the
space of the arguments in the second members in o* in which
the second members are analytic. We consider those solutions
of tf for which the indicated arguments lie in (.
The second members in oi may involve derivatives not in
the second members in a. The second members in oi will
be polynomials in the new derivatives, with coefficients
analytic in (.
Let w be the highest derivative present in a second member
in oi which is a derivative of a first member in oi. Then
iv is not present in any second member in <r, so that it
appears rationally and integrally in the second members in oi .
Let w be a derivative of v, the first member of the equation
v = g in oi . Then w can be replaced, in the second mem-
bers in oi, by its expression obtained on differentiating g.
We obtain thus a system ov with the same solutions as
oi (or o), and with the same first members as *i. The
system o- 2 satisfies condition (a). The derivatives higher than
w which appear in the second members in o* 2 also appear in
the second members in oi. Hence, if MI, present in the
second members in o" 2 , is a derivative of a first member in
0*2 then wi is lower than w. We treat w^ as w was treated.
Since there cannot be an infinite sequence of derivatives each
lower than the preceding one, we must arrive, in a finite
number of steps, at a system T, with the same solutions as cr,
which satisfies (a), (b), (c), and which has complete sets of
monomials corresponding to its first members. The second
members in T will be polynomials in any derivatives not
present in the second members of o-. Hence assumption (d)
is satisfied for ( and for any values of the new deri-
vatives. Thus r is orthonomic and has the same solutions
as o\*
* With .the values of the arguments in the second members in o lying
in .
146 ALGEBRAIC DIFFERENTIAL EQUATIONS
Of course, whether we employ o" or r, we get the same
set of principal derivatives and the same parametric deri-
vatives.
no. We consider an orthonomic system, or, whose first
members, as in 109, yield complete sets of monomials. We
are going to seek solutions of cr, analytic at some point,
which, with no loss of generality, may be taken as x% = 0,
i = 1 , , m .
Consider any yi. Let numerical values be assigned to the
parametric derivatives of yi, at the origin, with the sole
conditions that the second members in a are analytic for the
values given to the derivatives in them and that the series
where the a are the values of the parametric derivatives,
the subscripts indicating the type of differentiation, converges
in a neighborhood of the origin. The series (15) is called
the initial determination of y%. If yi does not appear in
a first member, (15) is a complete Taylor series.
In what follows, we suppose an initial determination to be
given for each yi. We shall then develop a process for
calculating the values of the principal derivatives at the
origin. There will result analytic functions yi which satisfy
each equation of a on the spread obtained by equating to
zero the non-multipliers of the monomial corresponding to the
first member. Later we shall obtain a condition for the yi
to give an actual solution of cr.
In the dissection (7) of each yi which we shall obtain,*
those terms whose monomials are multiples of monomials in
the complementary set will constitute the initial determination
of yi. Thus the initial determination of each yi is a linear
combination of a certain number of arbitrary functions, with
monomials for coefficients, the variables in the arbitrary
functions being specified. This description of the degree of
* This dissection is based on the complete set of monomials corre-
sponding to yi.
IX. RIQUIER'S EXISTENCE THEOREM 147
generality of the solution of a system of equations is one
of the most important aspects of Riquier's work.
We replace each yi which does not figure in any first
member in er by an arbitrarily selected initial determination.
Then a becomes an orthonomic system in the remaining yi,
with the same principal derivatives as before for the re-
maining yt. On this basis, we assume, with no loss of
generality, that every yt figures in a first member.
in. We use the symbol d to represent differential operators.
Any principal derivative, dy iy which is not a first member
in 0, can be obtained from one and only one first member
in er by differentiation with respect to multipliers of the
monomial corresponding to that first member. This is because
the first members yield complete sets. We have thus a unique
expression for dyi,
(16) d yi = ff,
where the derivatives in g are lower than Sy it
The infinite system obtained by adjoining all equations (16)
to (5 will be called r. Let p be any non-negative integer.
The systems of equations in T whose first members have p
for first mark will be called r p . Since the first mark of
a derivative is the sum of the order of the derivative and
of the first mark of the function differentiated, each T P has
only a finite number of equations.
Let a be the minimum, and b the maximum, of the first marks
in the first members in er. For the values assigned, in 110,
to the parametric derivatives, the equations r a ,r a +i, -, i^
determine uniquely the values at the origin of the principal
derivatives whose first mark does not exceed &. In short, the
lowest such derivative has an equation which determines it in
terms of parametric derivatives; the principal derivative next
in ascending order is determined in terms of parametric deri-
vatives, and, perhaps, the first principal derivative, and so on.
We subject the unknowns yj to the transformation
148 ALGEBRAIC DIFFERENTIAL EQUATIONS
where y; is the chosen initial determination of yj and where
the c are the principal derivatives at the origin of yj, of
first mark not exceeding 6, found as above.
Then a goes over into a system a' in the yj. In the new
system, we transpose the known terms in the first members
(these come from the known terms in (17)) to the right.
The new system will be orthonomic in the j/}, with the same
monomials for its first members as in o*. The second members
will be analytic when each Xi and each parametric derivative
is small.
The system T' for <r', analogous to r for <r, is obtained
by executing the transformation (17) on the equations of T.
Thus, if we give to the l/i, in #', initial determinations
which are identically zero, the principal derivatives at the
origin, of first mark not exceeding 6, will be determined as
zero by r f aj - . ., r' b .
On this account, we limit ourselves, without loss of
generality, to the search of solutions y^, , y n , of <r,
with initial determinations identically zero, assuming that
the system r a , , r b yields zero values at the origin for
the principal derivatives whose first marks do not exceed b.
112. In the second members in 374-1, no derivatives appear
whose first marks exceed 6 + 1. Those derivatives whose
first marks are 6 + 1 enter linearly, because they come from
the differentiation of derivatives of first mark 6 in r&.
We denote by d k yi the second member of (12). Then
every equation in ra+i is of the form
(18) Sty* = 2piajpdjyp + qi<*
where the dj yp are of first mark 6 + 1 and where the p and q
involve the x\ and derivatives whose first marks are b or less.
In (18), we consider every derivative of first mark 6 + 1
which is lower than di y tt to be present in the second member.
If necessary, we take p^p = .
Consider any diy a in (18). Suppose that there is a ft such
that yp has derivatives of first mark 6 + 1 which are lower
than <Jty For every such ft, we let rtap represent the
IX. RIQUIER'S EXISTENCE THEOREM 149
number of derivatives of yp, of first mark & + 1, which are
lower than <J z i/a. For every other fi, we let r ta/ j = 1, and
we suppose that a single derivative of yp of first mark
& + 1 appears in the second member of (18), with a zero
coefficient. We can thus not continue to say that every
derivative in the second member of (18) is lower than the
first member, but no difficulty will arise out of this; only
a question of language is involved.
Let r be the maximum of the n a p.
The p and q in (18) are analytic for small values of their
arguments. Let the p and q be expanded as series of powers
of their arguments.
Let > be such that each of the above series converges
for values of its arguments which all exceed e in modulus.
Let h > be such that each p and each q has a modulus
less than h when the arguments do not exceed c.
Let A be any positive number less than l/n.
Following 107, we determine positive numbers ?*, &, not
less than unity such that, if 8i y a and dj yp are of first mark
> with diycc higher than djyp, we have
In what follows, we associate with each yt a new unknown
function m.
Let
where 2 ranges over all derivatives of i^ , , u n whose
first mark does not exceed b (8iU is supposed to have the
same marks as <^y).
We consider the system of equations
(20)
,
150 ALGEBRAIC DIFFERENTIAL EQUATIONS
which has the general form of (18), with alterations of the
form of the p and q.
The function
h
is a majorant for every p and every q. As each ? is at
least unity, the same is true of h/(l e).
Thus, in virtue of (19), wherever a dj-yp is lower than diy n
in an equation in (18), the coefficient of djiip in the corre-
sponding equation of (20) will be a majorant for the coeffi-
cient of 8jyp. In the exceptional case where a djyp is not
lower thaji <5i?/ and thus has a zero coefficient, the corre-
sponding coefficient in (20) is certainly a majorant. Evidently
the terms in (20) which correspond to the q in (18) are
majorants of the q.
113. We shall show that (20) has a solution in which
each in is a function of
(21) ?l *l+"-
Consider, in (20), all derivatives of a particular U K whose
first marks are I + 1 . The first mark of any such derivative
is the order (total) of the derivative, plus the first mark of w.
Hence all of the derivatives of u a which are of first mark
&+1 are of the same order, say #.
Let the ?(, in what follows, represent functions of (21).
Put ua = ?4 and let Uai be the (i + + Oth derivative
of u'a with respect to (21). The)
8 V" lm u _ g<i . . . $<* t M '
8^..-a^ a ~ x ^^ M -'
1 7t
When the ?c are functions of (21), Q becomes a function ?'
of (21) and of the derivatives of the u n of order less than
gcc, = 1 , ., n. Equations (20) reduce to
(22) u* =
IX. RIQUIER'S EXISTENCE THEOREM
151
There will be n equations in (22), one for each . All
equations in (20) in which a given ?< appen v s in the first
member yield the same equation (22). We write (22) as
n
/t)o\ f i f [ 2 ^sr* ' i T
ffa * 0a p^ fffl
When (21) is zero and when the ui, i = 0, -, y tt 1 , for
each , are given zero values, the determinant of (22) with
respect to the n'ag a is
1 / , f* , , A
i, 1 A, .... I
A j "~ A . , I A
This determinant is not zero. In short, the equations
Z n = C H
(24)
imply
so that the determinant cannot vanish for
Then the u KQoL can be expressed as functions of the other
quantities in (23), analytic when the arguments are small.
By the existence theorem for ordinary differential equations,
(23) has a solution with the u tt i zero, for ?'<#, when (21)
is zero. The functions in this solution will be analytic for
(21) small.
114. We shall prove that, in the solution just found, all
u'ai with i ;> g a are positive for (21) zero. For (21) zero,
we have
* For i = 1 , subtract each equation from the first, in succession, and
substitute the results into the first.
152 ALGEBRAIC DIFFERENTIAL EQUATIONS
Referring to (24), we see that, since A<l/w, the zi are
positive if the a are all positive. Then the u' tt g a are positive
for every .
Differentiating (23), we find, for (21) zero,
where the k l( are positive. Again, the solution consists of
positive numbers. Continuing, we obtain our result.
What precedes shows that (20) has a solution, analytic at
the origin, with every derivative of first mark less than l-\-l
equal to zero and every other derivative positive, at the origin.
115. We now return to the system a. With the procedure
employed, in 111, for the determination, at the origin, of
the principal derivatives of first mark not greater than ft,
we determine the values of all principal derivatives at the
origin. We can ascend, step by step, through all the principal
derivatives, because each T P in 111 has only a finite number
of equations.
We obtain thus a complete power series for each yi. We
are going to prove that these power series converge for small
values of the ;r;.
Let dtt/u be any principal derivative. AVe shall prove that
the modulus of this derivative at the origin does not exceed
the value at the origin found for ^?e in 114.
For derivatives of first mark less than b + 1 , this is certainly
true; those derivatives have zero values. Let the result hold
for all derivatives lower than some diy y of first mark greater
than ft. The equation in r for d^j y is either in (18), or is
found by differentiating some equation in (1 8). Consider the
corresponding equation for <5/?< ; ,, which is either in (20), or
obtained from (20) by differentiation.
AVe shall consider the expressions for diy y and dtn y as
power series in the xi and in the derivatives in terms of
which 6,y y and d t n y are expressed.
AVe see that, for every term in the series for <J,-j/ y , there
is a dominating term in the series for di*t y . What is more,
IX. RIQUIER'S EXISTENCE THEOREM 153
the series for ditty may have other terms, involving d^iy
itself, or even higher derivatives. This is because of the
exceptional terms in (20), introduced in 112.*
Each term in di u r which has a corresponding term in dj y y
is at least as great as the modulus of that term at the origin,
for such terms involve only lower derivatives than dnjy or
ditty. Terms in d t uy which have no corresponding terms in
di y y are zero or positive at the origin. They will be positive
if they involve no #$, and contain only derivatives of first
mark at least b~\-l ( 114). This proves that the value
determined for each 6iy a by z has a modulus not greater
than the value at the origin of dn< a .
Thus the series obtained for the /// converge in a neigh-
borhood of the origin.
116. We shall now see to what extent the analytic func-
tions ytj just obtained, are solutions of a*.
Consider any equation dyi = y in tf. This equation, and
all equations obtained from it by differentiation with respect
to multipliers of the monomial corresponding to the lirst member,
are satisfied, at the origin, by the derivatives of 2A , , Un
at the origin. Hence, if we substitute ?A ,-,*/ into dy { #,
we obtain a function k of x\ , , x n which vanishes at the
origin, together with its derivatives with respect to the above
multipliers. Thus, in the expansion of k, only non-multipliers
occur. Then Jc vanishes when the non-multipliers are zero.
Hence y , - , y n satisfy each equation of G on the spread
obtained by equating to zero the non-multipliers corresponding
to the first member of the equation.
117. Let us return now to the most general orthonomic
system tf whose first members give complete sets of monomials.
We do not suppose that every y\ appears in some first member.
We consider any point xt = en, i = 1, , m, subject
to obvious conditions of analyticity. Let any values be given
to the parametric derivatives of the yi at #1, , a m ? so as
to yield convergent initial determinations. Then the principal
* In our present language, all derivatives in a second member in (18)
are lower than the first member.
154 ALGEBRAIC DIFFERENTIAL EQUATIONS
derivatives are determined uniquely by a in such a way as
to yield analytic functions 2/1, -,?/>* which satisfy each
equation in cr on the spread obtained by equating to m each
non-multiplier ^-corresponding to the first member of the equation.
This is an immediate consequence of the preceding sections.
PASSIVE OUTHONOMIC SYSTEMS
118. Let rf be an orthonomic system, described as in the
preceding section. Let the equations in # be listed so that
their first members form an ascending sequence, and let them
be written
(25) v t = 0, * = 1, , f.
If i'i is 8yj #, we attribute to vt the s marks of 8ijj. This
establishes order relations among the vi, according to the
convention of 106. To all of the derivatives of tv, we
attribute marks as in 106. Thus, the marks of dvi will
be the marks of the highest derivative in dn. By the
monomial corresponding to Vi, we mean the monomial corre-
sponding to 8yj. We shall refer to dyj as the first term
in vi. By the first term of a derivative of vi, we shall mean
the corresponding derivative of dyj.
Consider a v whose corresponding monomial, , has non-
multipliers. Let xi be such a non-multiplier. By 103,
Xi<* is the product of a ft, in the same complete set as
and higher than a, by multipliers of ft. Hence, there is a v pj
higher than v, such that some 8v p has the same first term
as dv/dxi. Then, in the expression
(26) _*
all derivatives effectively present are lower than the first
term of dv/dxi.
It is clear that (26) is a polynomial in such principal
derivatives as it may involve. Let w be the highest such
principal derivative. Then w is the first term of some ex-
pression 8v qy where dv q is lower than dv/dxi. We choose
Vq so that w is obtained from it by differentiation with respect
IX. RIQUIER'S EXISTENCE THEOREM loo
to multipliers of the corresponding monomial. Thi* makes
v q unique. Let then, identically,
(27) w = dv q + k,
where the derivatives in k are all lower than ic. We replace
w in (26) by its expression in (27) and find, identically.
-7j^7 = &Vp + hi(8v q , - - ),
where hi is a polynomial in 8v q whose coefficients involve
no principal derivative as high as iv. Let u\ be the highest
principal derivative in hi . We give it the treatment accorded
to w and find
where /^ is a polynomial in dv (1 , <Jtv- Continuing, we find
in a unique manner, an identity
(28) -~~ = cH^
in w T hich the coefficients in h involve only parametric deri-
vatives. We now write (28) in the form
(29) ~
where /* is the term of zero degree in h. Then i*> is an
expression in the parametric derivatives alone. The ex-
pression y vanishes when dv q , , 8v z are replaced by 0.
It is clear that, for any solution of </, we must have /* = 0.
The totality of equations ^ = 0, obtained from all equations
of cr for which the monomial corresponding to the first member
has non-multipliers, all non-multipliers being used, are called
the integr ability conditions for er.
119- If every expression ^ is identically zero, the system cr
is said to be passive.
We shall prove that, if <r is passive, the n functions
2/i, > yn, described in 117, which satisfy each equation
in o- on a certain spread, constitute an actual solution of #.
What we have to show is, that for these functions, every vt
in (25) vanishes identically.
156 ALGEBRAIC DIFFERENTIAL EQUATIONS
When the yj above are substituted into r/, we obtain
a function m of y\ , , x m . If n has no non-multipliers,
lit = 0. Otherwise, m vanishes when the non-multipliers
of the monomial corresponding* to ?v are equated to their /.
If, in (29), where /* is now identically zero, the parametric
derivatives in / are replaced by their expressions as functions
of the Xij found from the y/ z , (29) becomes a system y of
differential equations in the unknowns vi. Since (29) consisted
of identities, before these replacements, y is satisfied by
t'i = id, /=!,,.
We now attribute to each jr t an additional mark 0, and
to each Vi an additional mark t i. With this change, the
derivatives of the ?; t - will be completely ordered and the first
member in each equation in y will be higher than every
derivative in the second member.
If the second members in y contain derivatives of the first
members, we can get rid of such derivatives, step by step.
Then y goes over into an orthonomic system i/S with the
same first members as y.
For our purposes, it is unnecessary to adjoin new equations
to iff as in 109. Consider any unknown VL which appears
in a first member. The derivatives of r t in the first members
will be taken with respect to certain variables
(30) X a , ' ' ', *d-
The variables (30) when equated to their a/, give a spread on
which m vanishes.
The parametric derivatives of vt will be the derivatives
taken with respect to the variables not in (30). For the
corresponding m, each of these parametric derivatives is zero.
Now we know that, for given values of the parametric
derivatives, there is at most one solution of */>. But vi 0,
i = i J . . . ^ t is a solution of V f or which all parametric
derivatives vanish. Hence m = 0, i 1, , t.
This proves that, given a passive orthonomic system, there
is one and only one solution of the system for any given initial
determinations.
CHAPTER X
SYSTEMS OF ALGEBRAIC PARTIAL DIFFERENTIAL EQUATIONS
DECOMPOSITION OF A SYSTEM INTO IRREDUCIBLE SYSTEMS
1 20. We consider n unknown functions, ?/ 1? , y n , of m
independent variables, xi,-'-,x m . Definitions will usually
be as for the case of one independent variable, and will be
given, formally, only when there is some necessity for it.
We assume marks to have been assigned to the xi and yt
in such a way as to order completely the derivatives of
the y it
By a form, we shall mean a polynomial in the yi and any
number of their partial derivatives, with coefficients which
are functions of the Xi, meromorphic at each point of a given
open region 21 in the space of the xi. By a field, we shall
mean a set of functions meromorphic at each point of 21.
the set being closed with respect to rational operations and
partial differentiation. We assume a field & to be given in
advance. Where the contrary is not stated, the coefficients
in a form will belong to r.
121. By the leader of a form A which actually involves
unknowns, we shall mean the highest derivative present
in A.
Let A l and A 2 be two forms which actually involve unknowns.
If A* has a higher leader than A i7 then A s will be said to
be of higher rank than AI. If A and A 9 have the same
leader, and if the degree of A 2 in the common leader exceeds
that of Ai , then again, A 2 will be said to be of higher rank
than Ai. A form which effectively involves unknowns will
be said to be of higher rank than a form which does not.
157
158 ALGEBRAIC DIFFERENTIAL EQUATIONS
Two forms, for which no difference in rank is created by
what precedes, will be said to be of the same rank.*
The lemma of 2 goes over immediately to the case of
several variables.
122. If AI involves unknowns, A* will be said to be
reduced with respect to AI if A$ contains no derivative (proper)
of the leader of AI and if A^ is of lower degree than AI in
the leader of AI. A set of forms
(1) A 19 A t , ..-, A r
will be called an ascending set if either
(a) r = I and A v ^ 0, or
(b) r>l, AI involves unknowns and, for j>i, Aj is of higher
rank than At and reduced with respect to At.
When (b) holds, the leader of Aj is higher than that of A,-
for j>i.
Relative rank for ascending sets is defined exactly as in
3. If <Z>!, <Z> 2 , 0>s are ascending sets with <Z>i><Z> 2 and
2> ( I ) 3 then (P 1 >CD 8 . We prove the following lemma.
LEMMA. Let
(2) </>!, 0> 2 , -.., q , ...
le an infinite sequence of ascending sets such that <P 9 +i is not
higher than 0> q for any q. Then there exists a subscript r
such that, for q>r, O q has the same rank as <D r *
For q large, the first forms in the <D q have the same rank.
These first forms will either be free of the unknowns, or
else will have the same leader, say p r We have only to
consider the latter possibility, and may limit ourselves to
the case in which q with q large has at least two forms.
For q large, the second forms will have the same leader,
say p . Now p, 2 is not a proper derivative of p^ As we
saw above, p 2 is higher than p r We may confine ourselves
now to the case in which <Z> 7 , for q large, has at least three
* It will be noticed that the above definitions of relative rank do not
specialize into those of 2. This is due to the fact that the first mark
of each x,- is unity.
X. PARTIAL DIFFERENTIAL EQUATIONS 159
forms. Then, for q large, the third forms will all have the
same leader, j? 8 , higher than p l and p 2 and not a derivative
of either of them. Thus, our result holds, unless there is
an infinite sequence
Pi> P*> > P q > '"
of derivatives which increase steadily in rank, no p being
a derivative of a p i with i<q. But this contradicts Kiquier's
theorem on sequences of monomials proved in 99.
On the basis of the above lemma, we define a basic set
of a system 2 which contains non-zero forms, to be an
ascending set of 2 of least rank.
If AI in (1) involves unknowns, a form F will be said to
be reduced with respect to (1) it F is reduced with respect
to Ai, i = 1, -, r.
Let 2 be a system for which (1), with A t not free of the
unknowns, is a basic set. Then no non-zero form of 2 can
be reduced with respect to (1). If a non-zero form, reduced
with respect to (1), is adjoined to 2, the basic sets of the
resulting system are lower than (1).
123. In this section, we deal with an ascending set (1)
in which A involves unknowns.
If a form has a leader, p, we shall call the form 5 G/dp
the separant of G. The coefficient of the highest power of p
in G will be called the initial of G.
Let Si and It be, respectively, the separant and initial
of Ai in (1).
We shall prove the following result.
Let G be any form. There exist non-negative integers, $i,
ti, i = 1, , r, such that, when a suitable linear combination
of the Ai and a certain number of their derivatives, with forms
jor coefficients, is subtracted from
the remainder, R, is reduced with respect to (1).
Let pi be the leader of Ai. We limit ourselves, as we
may, to the case in which G involves derivatives, proper or
160 ALGEBRAIC DIFFERENTIAL EQUATIONS
improper, of the pi. Let the highest derivative in G which
is a derivative of a pi be q and let q be a derivative of PJ.
For the sake of uniqueness, if there are several possibilities
for;;, we use the largest ; available. To fix our ideas, we
assume q higher than p r . Then
SfG = CA$ + B
where Aj is a derivative of Aj with q for leader, and where
B is free of q. Because Aj and Sj involve no derivative
higher than q, B involves no derivative of a pi which is as
high as q. For uniqueness we take g as small as possible.
If B involves a derivative of a pi which is higher than # r ,
we give B the treatment accorded to G. After a finite
number of steps we arrive at a unique form D which differs
by a linear combination of derivatives of the Ai from a form
The form D involves no derivative of a pt which is higher
than p r .
We then find a relation
where K is reduced with respect to A r . The form K may
involve p r . Aside from p r , the only derivatives of the pi
present in K are derivatives otpi> -, pr-i* Such derivatives
are lower than p r . Let q^ be the highest of them.
Suppose that q^ is higher than jvi. We give K the
treatment accorded to G. In a finite number of steps, we
arrive at a unique form L which differs from some
by a linear combination of A r \ and the derivatives of
Ai, , A r -i. The form L is reduced with respect to A r
and A?-*. Aside from p r and jv-i, the derivatives of the
Pi in L are derivatives of pi, , jpr~2, and all such derivatives
are lower than
X. PARTIAL DIFFERENTIAL EQUATIONS 161
Continuing, we determine, in a unique, manner, a form R
as described in the statement of the lemma. We call R the
remainder of G with respect to (1).
124. The argument of 7-10 now goes over, without
change, to the case of several independent variables. We
secure the lemma:
LEMMA. Every infinite system of forms in y\, , y n has
a finite subsystem whose manifold is identical with that of the
infinite system*
As in 13, we prove the
THEOREM. Every system of forms in y^, , y n is equivalent
to a finite number of irreducible systems.
The decomposition is unique in the sense of 14.
As an example, we consider the equation
(3) z-(px-\-qy)-\-p* + q* = 0,
where p = dz/dx, q = dz/dy. Differentiating with respect
to a;, we find
(4) (rx + sy) + 2(pr + qs)
where r = d*z/dx 2 , s = d*z/dxdy, t = d*z/dy*. Differ-
entiating (3) with respect to y, we find
(5) (8x+ty) + 2(ps + qt) 0.
* This lemma is very different from, and is not to be confused with, the
theorem of Tresse for general (non- algebraic) systems of partial differential
equations. (Acta Mathematica, vol. 18, (1894), p. 4.) In using the implicit
function theorem to solve his system for certain derivatives, Tresse has
necessarily to confine himself to a portion of the manifold of his system.
In fact, it is not easy to imagine systems other than linear systems for
which Tresse's argument and result have a definite meaning. On this
basis, the above lemma, together with the theorem of 129, may be
regarded as an extension, to general algebraic systems, of Tresse's result,
as applied to linear systems. Thus the relation between Tresse's theorem
and our lemma is quite like that between the theorem that a system of
linear functions of n variables contains at most n + 1 linearly independent
functions, and Hilbert's theorem on th existence of a finite basis for any
system of polynomials in n variables. The use of Riquier's theorem of
99 was suggested to us by what is contained in Tresse's work. This
is the only common feature of the two arguments.
162 ALGEBRAIC DIFFERENTIAL EQUATIONS
From (4) and (5), we obtain
(rt s 2 )(x 2p) = 0; (rt - s 2 ) (y - 2q) = 0.
Thus, either rt s* = or z = (# 2 + ?/)/4. The latter
solution of (3) does not annul rt s 2 . Thus (3) is a reducible
system. As one can see from what follows, it is equivalent
to two irreducible systems.
BASIC SETS OF CLOSED IRREDUCIBLE SYSTEMS
125. Let 2 be a non-trivial closed irreducible system for
which
(6) A l9 Ai,-.-,A r ,
is a basic set. A solution of (6) for which no separant or
initial vanishes will be called a regular solution of (6).
Evidently such regular solutions exist. The remainder, with
respect to (6), of any form of 2. is zero. Hence, every regular
solution of (6) is a solution of 2. Furthermore, 2 consists
of all forms which vanish for the regular solutions of (6).
We represent by &, , m , more briefly by , a point
in 21 at which the coefficients in (6) are analytic. We use
the symbol fa] to designate any set of numerical values which
one may choose to associate with the derivatives appearing
in (6). The existence of regular solutions of (6) guarantees
the existence of a set , fa] for which every At vanishes,
but for which no separant or initial vanishes. In what
follows, we deal with such a set.
Let pi be the leader of Ai. The equation A t = 0, treated
as an algebraic equation for pi , determines pi as a function
of the xi and the derivatives lower than pi in A x , the
function being analytic for xt close to & and for the deri-
vatives lower than p t close to their values among the fa].
The value of the function pi for the special arguments
stipulated above will be the value for p^ in fa] . Let the
expression for pi be substituted into A 2 . We can then solve
A% for pz, expressing jp 2 as a function of the x% and of
X. PARTIAL DIFFERENTIAL EQUATIONS
the derivatives other than p t and p% appearing in A L and A t .
We substitute the expressions for pi and p 2 into A*,
solve 4 3 = for p s , and continue, in this manner, for all
forms in (6).
We will find thus a set of expressions for the pi, each
Pi being given as an analytic function of the Xi and of the
derivatives other than p L9 , p r in (6). We write
(7) pi = gij i = 1, , r.
If the equations (7) are considered as differential equations
for the yt, they will form an orthonomic system. We shall
prove the
THEOREM: The orthonomic system m = g\ is passive.
As in 65, we see that if (6) is considered as a set of
simple forms in the symbols for the derivatives, (6) will
be a basic set of a prime system, -//.* The unconditioned
unknowns in A will be those corresponding to the parametric
derivatives in (7). We form a simple resolvent for A, with
(8) W = 61 pi H h b r Prj
the bi being integers. Let the resolvent be
(9) Bouf+..- + B, = 0,
and let the expressions for the pt be
nm Ea+'-'+EitS-iU?- 1 - -,
(10) pt = - .p-' , a = 1, -, r,
(see 59), where the Bi, Eg and D are simple forms in the
(symbols for the) parametric derivatives. If the parametric
derivatives are specialized as functions of the Xi for which
B Q D^Q, the functions pi determined by (9) and (10) give
(in some open region) all of the solutions of A for the given
specialization of the parametric derivatives.
* All results relative to simple forms, which we employ, carry over
without difficulty to several variables.
164 ALGEBRAIC DIFFERENTIAL EQUATIONS
The relations (9) and (10) continue to hold if the pi are
replaced in (8) and (10) by the functions # appearing in (7).
For, let the arguments in the # be given any vulues, close
to those in , fy], for which B Q D does not vanish. Let
the value given to xt be JJ. If the parametric derivatives
are held fast at the values just assigned to them, while Xi
ranges over the neighborhood of , then the pi in (7) be-
come functions of the Xi, which, with the constant values
of the parametric derivatives, give a solution of A. For
this solution, B D ^ 0, so that (9) and (10) hold. Now the
values of the pi in this solution, at Xi = {, are the values
of the functions gt in (7) with the arguments specialized as
above. This shows that the gi can replace the pt in (8),
(9), (10).
We thus consider each gi in (7) to be expressed by the
second member of (10), where w is a function of the xi and
the parametric derivatives, analytic when the arguments are
close to their values in , [*/].
It may be, however, that the expressions (10) are meaning-
less for the particular values ?, [*/]; that is D may vanish
for those values. To take care of this point, and of a point
which will arise later, we pass to values ?', fa'], close to
, fy], for which DB Q K, where K is the discriminant of
(9), does not vanish. After we have proved the passivity
of (2) for the neighborhood of 5', [VL the passivity for the
neighborhood of , fyl will follow. *
Let equations be adjoined to (7), as in 109, so as to
form an orthonomic system, 0, whose first members give
complete sets of monomials. Let us see how the equations
in a can be written. From (10) we find
dpi _
(11)
dxj
(H iQ + + g^-i^- 1 ) + (Jo + + e/M
_
Let P represent the first member of (9). Let Q = dP/dw.
Then
X. PARTIAL DIFFERENTIAL EQUATIONS
(12) |^ = g. + ... + 01^
Now the resultant of P and Q equals, to within sign, B Q K.*
This means that
B Q K = LP+MQ
and that the denominator Q in (12) can be replaced by
B K (we multiply the numerator by M). In the new ex-
pression for dw/dxj, the degree of the numerator in w may
exceed s 1. We substitute this new expression for
into (11). Thus we have
(13)
T
where the Fi are free of w, and T is a product of powers
of D, B , K.
We get expressions similar to the second member of (13)
for all derivatives of the pi. If principal derivatives appear
in the Fi, we get rid of them, step by step. At the end,
we depress the degrees in w of the numerators in the ex-
pressions to less than s. This is accomplished by a division
by P; the division introduces a power of B Q into the deno-
minator.
All in all, each equation in <r will have the form
(14) dy = -^
T '
where T is a product of powers of D, B , K, where the F
are simple forms in the parametric derivatives.
If we refer now to 118, we see that every /* has for
the neighborhood of ', [*/], an expression like the second
member of (14). To establish the passivity of (7), for the
neighborhood of ?, fa], we have to show that every ^, as
a function of the xi and of the parametric derivatives, is
identically zero.
* Perron, Algebra, vol. 1, p. 225.
166 ALGEBRAIC DIFFERENTIAL EQUATIONS
The form DB<>K, which involves only parametric derivatives,
is reduced with respect to (6) and hence is not in 2. Con-
sider any regular solution of (6) for which DB K^Q.
Let i' , , ?m be a point at which DB Q K and the separants
and initials do not vanish, for this solution. Let [*/"] represent
the set of values, at ", - , ?m, for this solution, of the
derivatives in (6). Let us imagine that we have formed the
system (7) for the neighborhood of ?", IY']. Because the
calculation of the expressions for the fft, in (7), in terms
of w, involves only rational operations, the expressions will
be the same for ?", [i?"] as for ?', [*/]. The same is true
of the expressions for the p.
Suppose that the expression for some //>, say^, in terms
of w is not identically zero. Let Z be the numerator in
the expression for ^ . Then Z vanishes for the above solution
of (6). Because P is irreducible, the resultant W of P and Z
with respect to w is not identically zero. As W vanishes
for all solutions like the above, W is in 2. This contradicts
the fact that W involves only parametric derivatives.
Thus the expression for ^ is identically zero. Then ^i,
as an analytic function, vanishes for the neighborhood of
?', fa'].* As ', fy'] is arbitrarily close to , fy], ^ vanishes
for the neighborhood of , [^].
This proves the passivity of (7).
126. Let (6), with AI not free of the unknowns, be an
ascending set. We shall find necessary and sufficient con-
ditions for (6) to be a basic set for a closed irreducible
system.
As a first necessary condition, we have the condition
that (6), when regarded as a set of simple forms, be a basic
set for a prime system.
This implies the existence of r analytic functions gi, as
in (7), which annul the Ai, when substituted for the pi,
without annulling any initial or separant ( 45).
Let , fy] be some set of values, as in 125, for which
* Note that not all numbers in ' [y f ] are arguments of the /u .
X. PARTIAL DIFFERENTIAL EQUATIONS 167
no initial or separant vanishes. A second necessary con-
dition is that the system (7) be passive for the neighborhood
of Mil-
We shall prove that if (6), considered as a set of simple
forms, is a basic set of a prime system, and if (7) is passive
for a single set , fy], then (6) is a basic set of a closed ir-
reducible system.
Since (7) is passive for the neighborhood of , fy], the p
of 125 must vanish as analytic functions, for the neighbor-
hood of ?, [97]. Hence the expressions of the p in terms of w,
which are valid for the neighborhood of ?', [?'], vanish identi-
cally.
We conclude that for any set of values ?, [17] at all which
annul the AI but no initial or separant, (7) is passive.
The passivity of (7) for J, [ij\ as above implies that (6)
has regular solutions. We shall prove that the system 2 of
forms which vanish for all regular solutions of (6) is an ir-
reducible system of which (6) is a basic set.
Let O and H be such that OH is in -5". Let G and H^
be, respectively, the remainders of O and H with respect to
(6). There may be, in Oi and H lf parametric derivatives
not present in (6). But (6), considered as a set of simple
forms, will be the basic set of a prime system, 4, even after
the adjunction of the new parametric derivatives to the un-
knowns in the simple forms. Following 65, and using the
passivity established above, we see that every solution of A
which annuls no separant or initial in (6), leads, when con-
sidered at a quite arbitrary point of 21, to a regular solution
of (6). Thus GiHi, considered as a simple form, is in A*
Then one of G 1} H^ is in A. As in 65, it follows that
one of Gi,Hi vanishes identically. Then one of G, H is
in 2. Thus 2 is irreducible. What precedes shows that if
G is in 3, the remainder of G with respect to (6) is zero.
Then (6) is a basic set of 3.
127. Given a set (6) which satisfies the first condition of
126, we can determine with a finite number of differentiations,
rational operations and factorizations, whether or not (7) is
12
168 ALGEBRAIC DIFFERENTIAL EQUATIONS
passive. This follows from the fact that the expressions of
the /i in terms of w can be formed by a finite number of
such operations.
If (7) is not passive, the form WDB Q K, (as in 125),
which involves only parametric derivatives, vanishes for any
regular solutions which (6) may have.*
ALGORITHM FOR DECOMPOSITION
128. Let 2 be any finite system of forms, not all zero.
As in 67, we can get, by a finite number of differentiations,
rational operations and factorizations, a set, equivalent to 2,
of finite systems, -5*1, , 2 S , which have the following
properties:
(a) The basic sets of each 2i are not higher than those of -2;
(b) if the basic sets of 2i involve unknowns, the remainder
of any form of 2i with respect to a basic set is zero ;
(c) a basic set of 2i, considered as a set of simple forms,
is a basic set of a prime system.
Suppose that 2 t has a basic set (6), with A not free of
unknowns. If (7) is not passive, 2^ is equivalent to
where Si and Ii are the separant and initial of A*. Now
all of the latter systems have basic sets lower than (6).
If (7) proves passive, 2^ is equivalent to
where is the closed irreducible system of which (6) is
a basic set.
It is clear that by this process, we arrive, in a finite
number of steps, at a finite number of ascending sets, which
are basic sets of a set of irreducible systems equivalent to 2.
*It will be seen in 129 that WDB Q K vanishes for all solutions
which annul no initial.
X. PARTIAL DIFFERENTIAL EQUATIONS 169
The above constitutes a complete elimination theory for
systems of algebraic partial differential equations.
The test for a form to hold a system is as in 68.
One will notice that every system of linear partial differential
equations is irreducible.
ANALOGUE- OF THE HILBERT-NETTO THEOREM
129. We shall extend the theorem of 77 to the case of
several independent variables. As in the case of one variable,
it suffices to show that if the system
(15) Jl, Ft, ..-, Ft
has no solutions, then unity is a linear combination of the F{,
and of a certain number of their partial derivatives.
We suppose that unity has no such expression. One proves,
as in the case of one variable, that there is a point %,, a mj
at which the coefficients in (15) are analytic, for which certain
n power series
(16) Cot + Cn (Xi Oi) + + C m i (X m
render each Fj zero when substituted formally for y v , y n .
We shall use this fact to prove that (15) has analytic
solutions.
Let (15) be resolved into irreducible systems, by the
method of 128. Here, we are dealing with analytic solutions,
and not with formal ones. If we can show that one of the
irreducible systems has a basic set in which the first form
involves unknowns, we shall know that (15) has analytic
solutions.
Let us examine the process of decomposing (15) into ir-
reducible systems, following 128. First, it is apparent
that (16) is a formal solution of one of the systems ^.*
Let (16) be a solution of li. Let (6) be a basic set of 2 l .
*The coefficients in the 2 t may not be analytic at ai, , a*. In
that case, the coefficients are to be expressed as quotients of power series
for ai , , Om. This will be possible, since the coefficients are meromorphic.
170 ALGEBRAIC DIFFERENTIAL EQUATIONS
Then A iy in (6), involves unknowns. If the system (7) is
passive, then (15) has analytic solutions. Suppose that (7)
is not passive. We shall prove that
(17) li--Ir WDB Q K
vanishes for (16). Let us suppose that
I, .--Ir
does not vanish for (16).
The system of simple forms , obtained by adjoining the
simple form (see (8))
to A of 126, is indecomposable. We are dealing here with
analytic solutions of &. The forms
all hold &. Thus, if L is any one of the r + 1 forms (18),
L vanishes for jevery analytic solution of the system of simple
forms
(19) AI, , A r , W bipi ----- b r p r
for which /i I r does not vanish. By the Hilbert-Netto
theorem for simple forms, some power of
is a linear combination of the forms in (19). This means
that L vanishes for any formal power series solution of (19)
for which 1^ l r does not vanish.
Now, let the pi be series obtained by differentiating (16)
formally and let w be the series given by (8). We see that
w satisfies (9) and that the pi are given by (10).
If we go formally through the process of obtaining the (*
of 126, we find that the expression for every ^ in terms
X. PARTIAL DIFFERENTIAL EQUATIONS 171
of w vanishes for (16). Then, if /MI, for instance, is not
identically zero, W must vanish for (16).
Thus, if 2^ does not have a passive system (7), (1-6) is
a solution of one of the systems
Continuing, we find that (16) is a solution of a basic set of
some irreducible system 2' held by (15). Then the first form
of this basic set must involve unknowns, so that 2' has
analytic solutions.
This completes the proof of the analogue, for partial dif-
ferential forms, of the Hilbert-Netto theorem. It follows,
as in the case of one independent variable, that any finite
system of forms can be decomposed into finite irreducible
systems by differentiating the forms of tjie system a sufficient
number of times and resolving the extended system, considered
as a system of simple forms, into indecomposable systems.
INDEX
The numbers refer to sections
Adjunction of unknowns 17
arbitrary unknowns 24
ascending set 3
Basic set 4
Class of forms 2
complete set 102
Extended set 102
Field 1
fields of constants 18
form 2, 120
algebraically irreducible . . 19
general 34
quotients 38
simple 41
Hilbert-Netto theorem 77, 129
Indeterminate 34
initial 5, 123
integrability conditions 118
Jacobi-Weierstrass form 71
Kronecker's theorem 35
Leader 121
Luroth's theorem 91
Manifold 6
mark 106
monomial 99
multiple 99
multiplier 103
Open region 1
Parametric derivative 108
principal derivative 108
Rank 2, 121
relative reducibility 16
remainder 5, 123
resolvent 29
simple 42
resultant 34
Separant 5, 123
solution 6
general 19
normal 72
regular 20, 23, 125
seminormal 72
system
closed 23
complete 7
essential irreducible 14
indecomposable 41
irreducible 12
orthonomic 108
non-trivial 23
passive 119
prime 42
simply closed 41
systems equivalent 13
172
COLLOQUIUM PUBLICATIONS
Volume I. H. S. WHITE, Linear Systems of Curves on Algebraic
Surfaces; F. S. WOODS, Forms of Non-Euclidean Space;
E.B. VAN VLECK, Topics in the Theory of Divergent Series
and of Continued Fractions. 1905. 12 + 187 pp. $2.75.
Volume II. Out of Print.
Volume III. Out of Print.
Volume IV. L. E. DICKSON, On Invariants and the Theory of Numbers;
W. F. OSGOOD, Topics in the Theory of Functions of
Several Complex Variables. 1914. 4 + 230 pp. $2.50.
Volume V, PABT I. Out of print.
Volume V, PART II. OSWALD VEBLEN, Analysis Situs. Second edition.
1931. 10 + 194 pp. $2.00.
Volume VI. G. C. EVANS, The Logarithmic Potential. Discontinuous
Dirichlet and Neumann Problems. 1927. 8 + 150 pp. $2.00.
Volume VII. E. T. BELL, Algebraic Arithmetic. 1927. 4 + 180 pp. $2.50.
Volume VIII. L. P. EISENHART, Non-Riernannian Geometry. 1927.
8 + 184 pp. $2.50.
Volume IX. G. D. BIRKHOFF, Dynamical Systems. 1927. 8 + 295 pp.
$3.00.
Volume X. A. B. COBLE, Algebraic Geometry and Theta Functions.
1929. 8 + 282 pp. $3.00.
Volume XI. DUNHAM JACKSON, The Theory of Approximation. 1930.
8 + 178 pp. $2.50.
Volume XII. S. LEFSCHETZ, Topology. 1930. 10 + 410 pp. $4.50.
Volume XIII. R. L. MOORE, Foundations of Point Set Theory. 7 + 486 pp.
$5.00.
Volume XIV. J. F. RITT, Differential Equations from the Algebraic Stand-
point. 1932. 10 + 172 pp. $2.50.
Orders may be addressed to
AMERICAN MATHEMATICAL SOCIETY
501 West 116th Street
NEW YOBK CITY
or to any of the following official agents of the Society:
BOWES & BOWES, 1 and 2 Trinity Street, Cambridge, England ;
HIBSCHWALDSCHE BucHHANDLUNG, Unter den Linden 68, Berlin NW 7,
Germany ;
LIBEAIEIE SCIENTIFIQUE ALBERT BLANCH ABB, 3 Place de la Sorbonne,
Paris 5, France ;
LIBBEBIA EDITBICE NICOLA ZANICHELLI, 12, ViaA.de Togni, Milano(116),
Italic.
Live Music
Archive
Librivox
Free Audio
Metropolitan Museum
Cleveland
Museum of Art
Internet
Arcade
Console Living Room
Open Library
American
Libraries
TV News
Understanding
9/11